1 Introduction

Semantics: Adds numbers together.

See also Infix Operator "-" 6.4.3, Prefix Operator "+" 6.4.15

6.4.3 Infix Operator "-"

Summary: Subtract the second number from the first.

Syntax: Number Left - Number Right

Semantics: Subtracts one number from another number.

See also Infix Operator "+" 6.4.2, Prefix Operator "-" 6.4.16

6.4.4 Infix Operator "*"

Summary: Multiply two numbers.

Syntax: Number Left * Number Right

Semantics: Multiplies numbers together.

See also Infix Operator "+" 6.4.2, Infix Operator "/" 6.4.5

6.4.5 Infix Operator "/"

Summary: Divide the first number by the second.

Syntax: Number Left / Number Right

Semantics: Divides numbers. Dividing by zero returns an Error.

See also Infix Operator "-" 6.4.3, Infix Operator "*" 6.4.4

6.4.6 Infix Operator "^"

Summary: Exponentiation (Power).

Syntax: Number Left ^ Number Right

Constraints: NOT(AND(Left=0; Right=0)); Evaluators may evaluate expressions where OR(Left != 0; Right != 0) evaluates to a non-Error value.

Semantics: Returns POWER(Left, Right).

6.4.7 Infix Operator "="

Summary: Report if two values are equal

Syntax: Scalar Left = Scalar Right

Semantics: Returns TRUE if two values are equal. If the values differ in type, return FALSE. If the values are both Number, return TRUE if they are considered equal, else return FALSE. If they are both Text, return TRUE if the two values match, else return FALSE. For Text values, if the calculation setting HOST-CASE-SENSITIVE is false, text is compared but characters differencing only in case are considered equal. If they are both Logicals, return TRUE if they are identical, else return FALSE. Error values cannot be compared to a constant Error value to determine if that is the same Error value.

Evaluators may approximate and test equality of two numeric values with an accuracy of the magnitude of the given values scaled by the number of available bits in the mantissa, ignoring some least significant bits and thus providing compensation for not exactly representable values.

The result of “1=TRUE()” is FALSE for evaluators that implement a distinct Logical type and TRUE if they don't.

6.4.8 Infix Operator "<>"

Summary: Report if two values are not equal

Syntax: Any Left <> Any Right

Semantics: Returns NOT(Left = Right) if Left and Right are not Error. For Text values, if the calculation setting HOST-CASE-SENSITIVE is false, text is compared but characters differencing only in case are considered equal.

If either Left and Right are an Error, the result is an Error; this operator cannot be used to determine if two Errors are the same kind of Error.

Note: In some user interfaces the infix operator “<>” is displayed (or accepted) as “!=” or “≠”.

6.4.9 Infix Operator Ordered Comparison ("<", "<=", ">", ">=")

Summary: Report if two values have the given order

Syntax: Scalar Left op Scalar Right

where op is one of: "<", "<=", ">", or ">="

Semantics: Returns TRUE if the two values are less than, less than or equal, greater than, or greater than or equal (respectively). If both Left and Right are Numbers, compare them as numbers. If both Left and Right are Text, compare them as text; if the calculation setting HOST-CASE-SENSITIVE is false, text is compared but characters are compared ignoring case. If the values are both Logical, convert both to Number and then compare as Number.

These functions return one of TRUE, FALSE, or an Error if Left and Right have different types, but it is implementation-defined which of these results will be returned when the types differ.

See also Infix Operator "<>" 6.4.8, Infix Operator "=" 6.4.7

6.4.10 Infix Operator "&"

Summary: Concatenate two strings.

Syntax: Text Left & Text Right

Semantics: Concatenates two text (string) values.

Note: The infix operator “&” is equivalent to CONCATENATE(Left,Right).

6.4.11 Infix Operator Reference Range (":")

Summary: Computes an inclusive range given two references

Syntax: Reference Left : Reference Right

Semantics: Takes two references and computes the range, that is, a reference to the smallest 3-dimensional cube of cells that include both Left and Right including the cells on sheets positioned between Left and Right. Left and Right need not be a single cell. For an expression such as [.B4:.B5]:[.C5] the resulting range is B4:C5. In case Left and/or Right involve a reference list (result of operator reference union), the range is computed and extended for each element of the list(s).

Note: For example, (a,b,c,d denoting one reference each) (a~b):(c~d) computes a:b:c:d determining the outermost front-top-left and rear-bottom-right corners.

Left and Right may also be defined names or the result of a function returning a reference, such as INDIRECT.

See also Infix Operator Reference Union 6.4.13, Infix Operator Reference Intersection 6.4.12, INDIRECT 6.14.7

6.4.12 Infix Operator Reference Intersection ("!")

Summary: Compute the intersection of two references

Syntax: Reference Left ! Reference Right

Semantics: Takes two references and computes the intersection - a reference to the intersection of cells in both Left and Right. If there are no cells in common, returns an Error.

If Left or Right are not of type Reference or ReferenceList, an Error shall be returned.

If Left and/or Right are reference lists (result of infix operator reference concatenation), the intersection is computed for each combination of Left and Right, producing a reference list of intersections.

Note 1: For example (a,b,c,d denoting one reference each):
(a~b)!(c~d) will compute (a!c)~(a!d)~(b!c)~(b!d)

If for a resulting intersection there are no cells in common, the element is ignored and omitted from the result list. If for all intersections there are no cells in common and the result list is empty, an Error is returned.

Note 2: Intersection is notated as "!" in OpenFormula format, but as a space character in some user interfaces.

6.4.13 Infix Operator Reference Concatenation ("~") (aka Union)

Summary: Concatenate two references

Syntax: Reference Left ~ Reference Right

Returns: ReferenceList

Semantics: Takes two references and computes the "cell union", which is a concatenation of the reference Left followed by the reference Right. This is not the same as a union in set theory; duplicate references to cells are not removed. The resulting reference will have the number of areas, as reported by AREAS, as AREAS(Left)+AREAS(Right).

Note: Concatenation is notated as "~" in OpenFormula format, but as a comma or “+” in some user interfaces.

If Left or Right are not of type Reference or ReferenceList, an Error shall be returned.

Test Cases:

See also Infix Operator Reference Range 6.4.11, Infix Operator Reference Intersection 6.4.12, AREAS 6.13.2

6.4.14 Postfix Operator "%"

Summary: Divide the operand by 100

Syntax: Number Left %

Semantics: Computes Left / 100.

See also Prefix Operator "-" 6.4.16, Prefix Operator "+" 6.4.15

6.4.15 Prefix Operator "+"

Summary: No operation; returns its one argument.

Syntax: + Any Right

Semantics: Returns the value given to it. Note that this does not convert a value to the Number type. In fact, it does no conversion at all of a Number, Logical, or Text value - it returns the same Number, Logical, or Text value (respectively). The "+" applied to a reference may return the reference, or an Error.

6.4.16 Prefix Operator "-"

Summary: Negate its one argument.

Syntax: - Number Right

Semantics: Computes 0 - Right.

6.5 Matrix Functions

6.5.1 General

Matrix functions operate on matrices.

A matrix with M rows and N columns is defined by

A_{M \times N} = (\begin{matrix} a_{11} & a_{21} & \dots & a_{N 1} \\ a_{12} & a_{22} & \dots & a_{N 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 M} & a_{2 M} & \dots & a_{M N} \end{matrix})

The dimension subscript may be omitted, if the context allows it, i.e.

A_{M \times N} = A

. Matrices are represented by upper-case letters. The elements of a matrix are denoted by the corresponding lower case letter and subscripts, which defines the row and column number.

Square matrices have the same number of rows and columns, i.e.

M = N

6.5.2 MDETERM

Summary: Calculates the determinant of a matrix.

Syntax: MDETERM( ForceArray Array A )

Constraints: Only square matrices are allowed.

Semantics: Returns the determinant of matrix A. The determinant is defined by

\det (A_{N \times N}) = \sum_{P} sgn (P) \prod_{i = 1}^{N} a_{{ip}_{i}}

denotes a permutation of the numbers 1, 2, ..., n and

sgn (P)

is the sign of the permutation, which is +1 for an even amount of permutations (i.e., permutations that can be written as the composition of an even number of transpositions), -1 otherwise.

transposition on 1, ..., n is a permutation of 1, ..., n with exactly (n - 2) numbers fixed.

6.5.3 MINVERSE

Summary: Returns the inverse of a matrix.

Syntax: MINVERSE( ForceArray Array A )

Constraints: Only square matrices are allowed.

multiplied with its inverse

Calculates the inverse

A^{- 1}

of matrix

. The matrix

A^{- 1}

results in the unity matrix of the same dimension as

A_{N \times N} A_{N \times N}^{- 1} = A_{N \times N}^{- 1} A_{N \times N} = 1_{N \times N}

Invertible matrices have a non-zero determinant. If the matrix is not invertible, this function should return an Error value.

6.5.4 MMULT

Summary: Multiplies the matrices A and B.

Syntax: MMULT( ForceArray Array A ; ForceArray Array B )

Constraints: COLUMNS(A) = ROWS(B)

Returns the matrix product of the two matrices. The elements

c_{mn}

of the resulting matrix

C_{M \times N} = A_{M \times K} B_{K \times N}

, are defined by:

c_{mn} = \sum_{k = 1}^{K} a_{m k} b_{k n}

See also COLUMNS 6.13.5, ROWS 6.13.30

6.5.5 MUNIT

Summary: Creates a unit matrix of a specified dimension N.

Syntax: MUNIT( Integer N )

Constraints: The dimension has to be greater than zero.

Semantics: Creates the unit matrix (identity matrix) of dimension N.

1_{N \times N} = (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix})

6.5.6 TRANSPOSE

Summary: Returns the transpose of a matrix.

Syntax: TRANSPOSE( Array A )

Semantics: Returns the transpose AT of a matrix A, i.e. rows and columns of the matrix are exchanged.

A_{M \times N}^{T} = {(\begin{matrix} a_{11} & a_{21} & \dots & a_{M 1} \\ a_{12} & a_{22} & \dots & a_{M 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 N} & a_{2 N} & \dots & a_{M N} \end{matrix})}_{N \times M}

6.6 Bit operation functions

6.6.1 General

Evaluators shall support unsigned integer values and results of at least 48 bits (values from 0 to 2^48-1 inclusive). Operations that receive or result in a value that cannot be represented within 48 bits are implementation-defined.

6.6.2 BITAND

Summary: Returns bitwise “and” of its parameters

Syntax: BITAND( Integer X ; Integer Y )

Constraints: X ≥ 0, Y ≥ 0

Semantics: Returns bitwise “and” of its parameters. In the result, each bit position is 1 if and only if all parameters' bits at that position are also 1; else it is 0.

See also BITOR 6.6.4, BITXOR 6.6.6, AND 6.15.2

6.6.3 BITLSHIFT

Summary: Returns left shift of value X by N bits (“<<”)

Syntax: BITLSHIFT( Integer X ; Integer N )

Semantics: Returns left shift of value X by N bit positions:

Constraints: X ≥ 0

•If N < 0, return BITRSHIFT(X,-N)
•if N = 0, return X
•if N > 0, return X * 2^N

See also BITAND 6.6.2, BITXOR 6.6.6, BITRSHIFT 6.6.5

6.6.4 BITOR

Summary: Returns bitwise “or” of its parameters

Syntax: BITOR( Integer X ; Integer Y )

Constraints: X ≥ 0, Y ≥ 0

Semantics: Returns bitwise “or” of its parameters. In the result, each bit position is 1 if any of its parameters' bits at that position are also 1; else it is 0.

See also BITAND 6.6.2, BITXOR 6.6.6, AND 6.15.2

6.6.5 BITRSHIFT

Summary: Returns right shift of value X by N bits (“>>”)

Syntax: BITRSHIFT( Integer X ; Integer N )

Semantics: Returns right shift of value X by N bit positions:

Constraints: X ≥ 0

•If N < 0, return BITLSHIFT(X,-N)
•if N = 0, return X
•if N > 0, return INT(X / 2^N)

See also BITAND 6.6.2, BITXOR 6.6.6, BITLSHIFT 6.6.3, INT 6.17.2

6.6.6 BITXOR

Summary: Returns bitwise “exclusive or” of its parameters

Syntax: BITXOR( Integer X ; Integer Y )

Constraints: X ≥ 0, Y ≥ 0

Semantics: Returns bitwise “exclusive or” (xor) of its parameters. In the result, each bit position is 1 if one or the other parameters' bits at that position are 1; else it is 0.

See also BITAND 6.6.2, BITOR 6.6.4, OR 6.15.8

6.7 Byte-position text functions

6.7.1 General

Byte-position text functions are like their equivalent ordinary text functions, but manipulate byte positions rather than a count of the number of characters. Byte positions are integers that may depend on the specific text representation used by the implementation. Byte positions are by definition implementation-dependent and reliance upon them reduces interoperability.

The pseudotypes ByteLength and BytePosition are Integers, but their exact meanings and values are not further defined by this specification.

6.7.2 FINDB

Summary: Returns the starting position of a given text, using byte positions.

Syntax: FINDB( Text Search ; Text T [ ; BytePosition Start ] )

Returns: BytePosition

Semantics: The same as FIND, but using byte positions.

See also FIND 6.20.9 , LEFTB 6.7.3 , RIGHTB 6.7.7

6.7.3 LEFTB

Summary: Returns a selected number of text characters from the left, using a byte position.

Syntax: LEFTB( Text T [ ; ByteLength Length ] )

Semantics: As LEFT, but using a byte position.

See also LEFT 6.20.12, RIGHT 6.20.19, RIGHTB 6.7.7

6.7.4 LENB

Summary: Returns the length of given text in units compatible with byte positions

Syntax: LENB( Text T )

Returns: ByteLength

Semantics: As LEN, but compatible with byte position values.

See also LEN 6.20.13, LEFTB 6.7.3, RIGHTB 6.7.7

6.7.5 MIDB

Summary: Returns extracted text, given an original text, starting position using a byte position, and length.

Syntax: MIDB( Text T ; BytePosition Start ; ByteLength Length )

Semantics: As MID, but using byte positions.

Constraints: Length ≥ 0.

See also MID 6.20.15, LEFTB 6.7.3, RIGHTB 6.7.7, REPLACEB 6.7.6

6.7.6 REPLACEB

Summary: Returns text where an old text is replaced with a new text, using byte positions.

Syntax: REPLACEB( Text T ; BytePosition Start ; ByteLength Len ; Text New )

Semantics: As REPLACE, but using byte positions.

See also REPLACE 6.20.17, LEFTB 6.7.3, RIGHTB 6.7.7, MIDB 6.7.5, SUBSTITUTE 6.20.21

6.7.7 RIGHTB

Summary: Returns a selected number of text characters from the right, using byte position.

Syntax: RIGHTB( Text T [ ; ByteLength Length ] )

Semantics: As RIGHT, but using byte positions.

6.7.8 SEARCHB

Summary: Returns the starting position of a given text, using byte positions.

Syntax: SEARCHB( Text Search ; Text T [ ; BytePosition Start ] )

Returns: BytePosition

Semantics: As SEARCH, but using byte positions.

See also SEARCH 6.20.20, EXACT 6.20.8, FIND 6.20.9, FINDB 6.7.2

6.8 Complex Number Functions

6.8.1 General

Functions for complex numbers.

6.8.2 COMPLEX

Summary: Creates a complex number from a given real coefficient and imaginary coefficient.

Syntax: COMPLEX( Number Real ; Number Imaginary [ ; Text Suffix ] )

Semantics: Constructs a complex number from the given coefficients. The third parameter Suffix is optional, and should be either “i” or “j”. Upper case “I” or “J” are not accepted for the suffix parameter.

6.8.3 IMABS

Summary: Returns the absolute value of a complex number.

Syntax: IMABS( Complex X )

= a + bj, the absolute value =

\sqrt{a^{2} + b^{2}}

; if

= r(cos

+ isin

φ), the absolute value = r.

6.8.4 IMAGINARY

Summary: Returns the imaginary coefficient of a complex number.

Syntax: IMAGINARY( Complex X )

Semantics: If X = a + bi or X = a + bj, then the imaginary coefficient is b.

6.8.5 IMARGUMENT

Summary: Returns the complex argument of a complex number.

Syntax: IMARGUMENT( Complex X )

Semantics: If X = a + bi = r(cosφ + isinφ), a or b is not 0 and -π < φ ≤ π, then the complex argument is φ. φ is expressed by radians. If X = 0, then IMARGUMENT(X) is implementation-defined and either 0 or an error.

6.8.6 IMCONJUGATE

Summary: Returns the complex conjugate of a complex number.

Syntax: IMCONJUGATE( Complex X )

Semantics: If X = a + bi, then the complex conjugate is a - bi.

6.8.7 IMCOS

Summary: Returns the cosine of a complex number.

Syntax: IMCOS( Complex X )

Semantics: If X = a + bi, then cos(X) = cos(a)cosh(b) - sin(a)sinh(b)i.

6.8.8 IMCOSH

Summary: Returns the hyperbolic cosine of a complex number.

Syntax: IMCOSH( Complex N )

Semantics: If N = a + bi, then cosh(N) = cosh(a)cos(b) + sinh(a)sin(b)i.

6.8.9 IMCOT

Summary: Returns the cotangent of a complex number.

Syntax: IMCOT(Complex N)

Semantics: Equivalent to the following (except N is computed only once):

IMDIV(IMCOS(N);IMSIN(N))

See also IMCOS 6.8.7, IMDIV 6.8.12, IMSIN 6.8.20, IMTAN 6.8.27

6.8.10 IMCSC

Summary: Returns the cosecant of a complex number.

Syntax: IMCSC(Complex N)

Semantics: Equivalent to the following:

IMDIV(1;IMSIN(N))

6.8.11 IMCSCH

Summary: Returns the hyperbolic cosecant of a complex number.

Syntax: IMCSCH( Complex N )

Semantics: Computes the hyperbolic cosecant. This is equivalent to:

IMDIV(1;IMSINH(N))

6.8.12 IMDIV

Summary: Divides the first number by the second.

Syntax: IMDIV( Complex X ; Complex Y )

Semantics: Given X = a + bi and Y = c + di, return the quotient

\frac{(ac + bd) + (bc - ad) i}{(c^{2} + d^{2})}

Division by zero returns an Error.

6.8.13 IMEXP

Summary: Returns the exponent of e and a complex number.

Syntax: IMEXP( Complex X )

Summary: Returns the natural logarithm of a complex number.

= a + bi, the result is

e^{a} (\cos b + i \sin b)

6.8.14 IMLN

Syntax: IMLN( Complex X )

Semantics: COMPLEX(LN(IMABS(X)); IMARGUMENT(X)) .

See also COMPLEX 6.8.2, IMABS 6.8.3, IMARGUMENT 6.8.5, IMEXP 6.8.13 , IMLOG10 6.8.15, LN 6.16.39

6.8.15 IMLOG10

Summary: Returns the common logarithm of a complex number.

Syntax: IMLOG10( Complex X )

Semantics: IMLOG10(X) is IMDIV(IMLN(X);COMPLEX(LN(10);0)) .

See also COMPLEX 6.8.2, IMDIV 6.8.12, IMLN 6.8.14 , IMPOWER 6.8.17, LN 6.16.39

6.8.16 IMLOG2

Summary: Returns the binary logarithm of a complex number.

Syntax: IMLOG2( Complex X )

Semantics: IMLOG2(X) is IMDIV(IMLN(X);COMPLEX(LN(2);0)) .

See also COMPLEX 6.8.2, IMDIV 6.8.12, IMLN 6.8.14 , IMPOWER 6.8.17, LN 6.16.39

6.8.17 IMPOWER

Summary: Returns the complex number X raised to the Yth power.

Syntax: IMPOWER( Complex X ; Complex Y ) or IMPOWER( Complex X ; Number Y)

Semantics: IMPOWER(X;Y) is IMEXP(IMPRODUCT(Y; IMLN(X)))

An evaluator implementing this function shall permit any Number Y but may also allow any Complex Y.

See also IMEXP 6.8.13, IMLN 6.8.14, IMPOWER 6.8.17, IMPRODUCT 6.8.18

6.8.18 IMPRODUCT

Summary: Returns the product of complex numbers.

Syntax: IMPRODUCT( { ComplexSequence N }+ )

Semantics: Multiply the complex numbers together. Given two complex numbers X = a + bi and
Y = c + di, the product X * Y = (ac - bd) + (ad + bc)i

6.8.19 IMREAL

Summary: Returns the real coefficient of a complex number.

Syntax: IMREAL( Complex N )

Semantics: If N = a + bi or N = a + bj, then the real coefficient is a.

6.8.20 IMSIN

Summary: Returns the sine of a complex number.

Syntax: IMSIN( Complex N )

Semantics: If N = a + bi, then sin(N) = sin(a)cosh(b) + cos(a)sinh(b)i.

6.8.21 IMSINH

Summary: Returns the hyperbolic sine of a complex number.

Syntax: IMSINH( Complex N )

Semantics: If N = a + bi, then sinh(N) = sinh(a)cos(b) + cosh(a)sin(b)i.

6.8.22 I MSEC

Summary: Returns the secant of a complex number.

Syntax: IMSEC(Complex N )

Semantics: Equivalent to the following:

IMDIV(1;IMCOS(N))

6.8.23 IMSECH

Summary: Returns the hyperbolic secant of a complex number.

Syntax: IMSECH( Complex N )

Semantics: Computes the hyperbolic secant. This is equivalent to:

IMDIV(1;IMCOSH(N))

See also IMCOSH 6.8.8, IMDIV 6.8.12, SECH 6.16.57

6.8.24 IMSQRT

Summary: Returns the square root of a complex number.

Syntax: IMSQRT( Complex N )

Semantics: If N = 0 + 0i, then IMSQRT(N) = 0. Otherwise IMSQRT(N) is
SQRT(IMABS(N)) * sin(IMARGUMENT(N) / 2) + SQRT(IMABS(N)) * cos(IMARGUMENT(N) / 2)i.

See also IMABS 6.8.3, IMARGUMENT 6.8.5, IMPOWER 6.8.17, SQRT 6.16.58

6.8.25 IMSUB

Summary: Subtracts the second complex number from the first.

Syntax: IMSUB( Complex X ; Complex Y )

Semantics: Subtract complex number Y from X.

6.8.26 IMSUM

Summary: Sums (add) a set of complex numbers, including all numbers in ranges.

Syntax: IMSUM( { ComplexSequence N }+ )

Semantics: Adds complex numbers together. Text that cannot be converted to a complex number is ignored.

It is implementation-defined what happens if this function is given zero parameters; an evaluator may either produce an Error or the Number 0 if it is given zero parameters.

6.8.27 IMTAN

Summary: Returns the tangent of a complex number

Syntax: IMTAN(Complex N)

Semantics: Equivalent to the following (except N is computed only once):

IMDIV(IMSIN(N);IMCOS(N))

See also IMDIV 6.8.12, IMSIN 6.8.20, IMCOS 6.8.7, IMCOT 6.8.25

6.9 Database Functions

6.9.1 General

Database functions use the variables, Database 4.11.9, Field 4.11.10, and Criteria 4.11.11.

The results of database functions may change when the values of the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties change. 3.4

6.9.2 DAVERAGE

Summary: Finds the average of values in a given field from the records (rows) in a database that match a search criteria.

Syntax: DAVERAGE( Database D ; Field F ; Criteria C )

Semantics: Perform AVERAGE on data records in database D field F that match criteria C.

See also AVERAGE 6.18.3, COUNT 6.13.6, DSUM 6.9.11, DCOUNT 6.9.3, SUM 6.16.61

6.9.3 DCOUNT

Summary: Counts the number of records (rows) in a database that match a search criteria and contain numerical values.

Syntax: DCOUNT( Database D ; [ Field F ] ; Criteria C )

Semantics: Perform COUNT on data records in database D field F that match criteria C. If the Field argument is omitted, DCOUNT returns the count of all records that satisfy Criteria C.

See also COUNT 6.13.6, COUNTA 6.13.7, DCOUNTA 6.9.4, DSUM 6.9.11

6.9.4 DCOUNTA

Summary: Counts the number of records (rows) in a database that match a search criteria and contain values (as COUNTA).

Syntax: DCOUNTA( Database D ; [ Field F ] ; Criteria C )

Semantics: Perform COUNTA on data records in database D field F that match criteria C. If the Field argument is omitted, DCOUNTA returns the count of all records that satisfy criteria C.

See also COUNT 6.13.6, COUNTA 6.13.7, DCOUNT 6.9.3, DSUM 6.9.11

6.9.5 DGET

Summary: Gets the single value in the field from the single record (row) in a database that matches a search criteria.

Syntax: DGET( Database D ; Field F ; Criteria C )

Semantics: Extracts the value in field F of the one data record in database D that matches criteria C. If no records match, or more than one matches, it returns an Error.

See also DMAX 6.9.6, DMIN 6.9.7, DSUM 6.9.11

6.9.6 DMAX

Summary: Finds the maximum value in a given field from the records (rows) in a database that match a search criteria.

Syntax: DMAX( Database D ; Field F ; Criteria C )

Semantics: Perform MAX on only the data records in database D field F that match criteria C.

See also MAX 6.18.45, DMIN 6.9.7, MIN 6.18.48

6.9.7 DMIN

Summary: Finds the minimum value in a given field from the records (rows) in a database that match a search criteria.

Syntax: DMIN( Database D ; Field F ; Criteria C )

Semantics: Perform MIN on only the data records in database D field F that match criteria C.

See also MIN 6.18.48, DMAX 6.9.6, MAX 6.18.45

6.9.8 DPRODUCT

Summary: Finds the product of values in a given field from the records (rows) in a database that match a search criteria.

Syntax: DPRODUCT( Database D ; Field F ; Criteria C )

Semantics: Multiply together only the data records in database D field F that match criteria C.

6.9.9 DSTDEV

Summary: Finds the sample standard deviation in a given field from the records (rows) in a database that match a search criteria.

Syntax: DSTDEV( Database D ; Field F ; Criteria C )

Semantics: Perform STDEV on only the data records in database D field F that match criteria C.

6.9.10 DSTDEVP

Summary: Finds the population standard deviation in a given field from the records (rows) in a database that match a search criteria.

Syntax: DSTDEVP( Database D ; Field F ; Criteria C )

Semantics: Perform STDEVP on only the data records in database D field F that match criteria C.

6.9.11 DSUM

Summary: Finds the sum of values in a given field from the records (rows) in a database that match a search criteria.

Syntax: DSUM( Database D ; Field F ; Criteria C )

Semantics: Perform SUM on only the data records in database D field F that match criteria C.

See also SUM 6.16.61, DMIN 6.9.7, DMAX 6.9.6

6.9.12 DVAR

Summary: Finds the sample variance in a given field from the records (rows) in a database that match a search criteria.

Syntax: DVAR( Database D ; Field F ; Criteria C )

Semantics: Perform VAR on only the data records in database D field F that match criteria C.

6.9.13 DVARP

Summary: Finds the population variance in a given field from the records (rows) in a database that match a search criteria.

Syntax: DVARP( Database D ; Field F ; Criteria C )

Semantics: Perform VARP on only the data records in database D field F that match criteria C.

6.10 Date and Time Functions

6.10.1 General

6.10.2 DATE

Summary: Constructs a date from year, month, and day of month.

Syntax: DATE( Integer Year ; Integer Month ; Integer Day )

Constraints: 1904 ≤ Year ≤ 9956; 1 ≤ Month ≤ 12; 1 ≤ Day ≤ 31; Evaluators may evaluate expressions that do no meet this constraint.

Semantics: This computes the date's serial number given Year, Month, and Day of the Gregorian calendar. Fractional values are truncated. Month > 12 and Day > days of Month will roll over the date, computing the result by adding months and days as necessary. The value of the serial number depends on the current epoch.

6.10.3 DATEDIF

Summary: Returns the difference in years, months, or days of two date numbers.

Syntax: DATEDIF( DateParam StartDate ; DateParam EndDate ; Text Format )

Semantics: Compute difference of StartDate and EndDate, in the units given by Format.

The Format is a code from the following table, entered as text, that specifies the format you want the result of DATEDIF to have.

Table 5 - DATEDIF

Format	Returns the number of
y	Years
m	Months. If there is not a complete month between the dates, 0 will be returned.
d	Days
md	Days, ignoring months and years
ym	Months, ignoring years
yd	Days, ignoring years

See also DAYS360 6.10.7, DAYS 6.10.6, Infix Operator “-” 6.4.3

6.10.4 DATEVALUE

Summary: Returns the date serial number from given text.

Syntax: DATEVALUE( Text D )

Semantics: This computes the serial number of the text string D, using the current locale. This function shall accept ISO date format (YYYY-MM-DD), which is locale-independent. It is semantically equal to VALUE(Date), if Date has a date format, since text matching a date format is automatically converted to a serial number when used as a Number. If the text of D has a combined date and time format, e.g. YYYY-MM-DD HH:MM:SS, the integer part of the date serial number is returned. If the text of D does not have a date or time format, an evaluator may return an Error. See VALUE for more information on date formats. The value of the serial number depends on the current epoch.

See also TIME 6.10.17, DATE 6.10.2, TIMEVALUE 6.10.18, VALUE 6.13.34

6.10.5 DAY

Summary: Returns the day from a date.

Syntax: DAY( DateParam D )

Semantics: Returns the day portion of D.

6.10.6 DAYS

Summary: Returns the number of days between two dates

Syntax: DAYS( DateParam EndDate ; DateParam StartDate )

Semantics: Returns the number of days between two dates. If StartDate and EndDate are Numbers, this is EndDate – StartDate. If they are both Text, this is DATEVALUE(StartDate) – DATEVALUE(EndDate).

See also DATEDIF 6.10.3, DATEVALUE 6.10.4, DAYS360 6.10.7, MONTH 6.10.13, YEAR 6.10.23, Infix Operator “-” 6.4.3

6.10.7 DAYS360

Summary: Returns the number of days between two dates using the 360-day year

Syntax: DAYS360( DateParam StartDate ; DateParam EndDate [ ; Logical Method = FALSE ] )

Semantics: If Method is FALSE, it uses the National Association of Securities Dealers (NASD) method, also known as the U.S. method. If Method is TRUE, the European method is used.

The US/NASD method (30US/360):

1.Truncate date values, set sign = 1.
2.If StartDate's day-of-month is 31, it is changed to 30.
3.Otherwise, if StartDate's day-of-month is the last day of February, it is changed to 30.
4.If EndDate's day-of-month is 31 and StartDate's day-of-month is 30 (after having applied a change for #2 or #3, if necessary), EndDate's day-of-month is changed to 30.
Note 1: This calculation is slightly different from Basis 0 (4.11.7 Basis). Dates are never swapped.

The European method (30E/360):

1.Truncate date values, set sign = 1.
2.If StartDate is after EndDate then swap dates and set sign = -1.
3.If StartDate's day-of-month is 31, it is changed to 30.
4.If EndDate's day-of-month is 31, it is changed to 30.
Note 2: Days in February are never changed.
Note 3: This calculation is identical to Basis 4 (4.11.7 Basis)

For both methods the value then returned is
sign * ((EndDate.year * 360 + EndDate.month * 30 + EndDate.day) – (StartDate.year * 360 + StartDate.month * 30 + StartDate.day))

6.10.8 EDATE

Summary: Returns the serial number of a given date when MonthAdd months is added

Syntax: EDATE( DateParam StartDate ; Number MonthAdd )

Semantics: First truncate StartDate and MonthAdd, then add MonthAdd number of months. MonthAdd can be positive, negative, or 0; if zero, the number representing StartDate (in the current epoch) is returned.

If after adding the given number of months, the day of month in the new month is larger than the number of days in the given month, the day of month is adjusted to the last day of the new month. Then the serial number of that date is returned.

See also DAYS 6.10.6, DATEDIF 6.10.3, EOMONTH 6.10.9

6.10.9 EOMONTH

Summary: Returns the serial number of the end of a month, given date plus MonthAdd months

Syntax: EOMONTH( DateParam StartDate ; Integer MonthAdd

Semantics: First truncate StartDate and MonthAdd, then add MonthAdd number of months. MonthAdd can be positive, negative, or 0. Then return the serial number representing the end of that month. Due to the semantics of this function, the value of DAY(StartDate) is irrelevant.

6.10.10 HOUR

Summary: Extracts the hour (0 through 23) from a time.

Syntax: HOUR( TimeParam T )

Semantics: Extract from T the hour value, 0 through 23, as per a 24-hour clock. This is equal to:

DayFraction = (T - INT(T))

Hour = INT(DayFraction * 24)

See also MONTH 6.10.13, DAY 6.10.5, MINUTE 6.10.12, SECOND 6.10.16, INT 6.17.2

6.10.11 ISOWEEKNUM

Summary: Determines the ISO week number of the year for a given date.

Syntax: ISOWEEKNUM( DateParam D )

Summary: Extracts the minute (0 through 59) from a time.

Semantics: Returns the ordinal number of the [ISO8601] calendar week in the year for the given date D. ISO 8601 defines the calendar week as a time interval of seven calendar days starting with a Monday, and the first calendar week of a year as the one that includes the first Thursday of that year.

See also DAY 6.10.5, MONTH 6.10.13, YEAR 6.10.23, WEEKDAY 6.10.20, WEEKNUM 6.10.21

6.10.12 MINUTE

Syntax: MINUTE( TimeParam T )

Semantics: Extract from T the minute value, 0 through 59, as per a clock. This is equal to:

DayFraction = (T - INT(T))

HourFraction = (DayFraction * 24 - INT(DayFraction * 24))

Minute = INT(HourFraction * 60)

See also MONTH 6.10.13, DAY 6.10.5, HOUR 6.10.10, SECOND 6.10.16, INT 6.17.2

6.10.13 MONTH

Summary: Extracts the month from a date.

Syntax: MONTH( DateParam Date )

Semantics: Takes Date and returns the month portion.

6.10.14 NETWORKDAYS

Summary: Returns the whole number of work days between two dates.

Syntax: NETWORKDAYS( DateParam Date1 ; DateParam Date2 [ ; [ DateSequence Holidays ] [ ; LogicalSequence Workdays ] ] )

Semantics: Returns the whole number of work days between two dates.

Work days are defined as non-weekend, non-holiday days. By default, weekends are Saturdays and Sundays and there are no holidays.

The optional 3rd parameter Holidays can be used to specify a list of dates to be treated as holidays. Note that this parameter can be omitted as an empty parameter (two consecutive ;; semicolons) to be able to pass the set of Workdays without Holidays.

The optional 4th parameter Workdays can be used to specify a different definition for the standard work week by passing in a list of numbers which define which days of the week are workdays (indicated by 0) or not (indicated by non-zero) in order Sunday, Monday,...,Saturday. So, the default definition of the work week excludes Saturday and Sunday and is: {1;0;0;0;0;0;1}. To define the work week as excluding Friday and Saturday, the third parameter would be: {0;0;0;0;0;1;1}.

6.10.15 NOW

Summary: Returns the serial number of the current date and time.

Syntax: NOW()

Returns: DateTime

Semantics: This returns the current day and time serial number, using the current locale. If you want only the serial number of the current day, use TODAY 6.10.19.

See also DATE 6.10.2, TIME 6.10.17, TODAY 6.10.19

6.10.16 SECOND

Summary: Extracts the second (the integer 0 through 59) from a time. This function presumes that leap seconds never exist.

Syntax: SECOND( TimeParam T )

Semantics: Extract from T the second value, 0 through 59, as per a clock. Note that this returns an integer, without a fractional part. Note also that this rounds to the nearest second, instead of returning the integer part of the seconds. This is equal to:

DayFraction = (T - INT(T))

HourFraction = (DayFraction * 24 - INT(DayFraction * 24))

MinuteFraction = (HourFraction * 60 - INT(HourFraction * 60))

Second = ROUND(MinuteFraction * 60)

See also MONTH 6.10.13, DAY 6.10.5, HOUR 6.10.10, MINUTE 6.10.12, INT 6.17.2

6.10.17 TIME

Summary: Constructs a time value from hours, minutes, and seconds.

Syntax: TIME( Number Hours ; Number Minutes ; Number Seconds )

Returns: Time

Constraints: None. Evaluators may first perform INT() on the hour, minute, and second before doing the calculation.

Semantics: Returns the fraction of the day consumed by the given time, i.e.:

((Hours * 60 * 60) + (Minutes * 60) + Seconds) / (24 * 60 * 60)

Time is a subtype of Number, where a time value of 1 = 1 day = 24 hours.

Hours, Minutes, and Seconds may be any number (they shall not be limited to the ranges 0..24, 0..59, or 0..60 respectively).

6.10.18 TIMEVALUE

Summary: Returns a time serial number from given text.

Syntax: TIMEVALUE( Text T )

Returns: Time

Semantics: This computes the serial number of the text string T, which is a time, using the current locale. This function shall accept ISO time format (HH:MM:SS), which is locale-independent. If the text of T has a combined date and time format, e.g. YYYY-MM-DD HH:MM:SS, the fractional part of the date serial number is returned. If the text of T does not have a time format, an evaluator may attempt to convert the number another way (e.g., using VALUE), or it may return an Error (this is implementation-dependent).

See also TIME 6.10.17, DATE 6.10.2, DATEVALUE 6.10.4, VALUE 6.13.34

6.10.19 TODAY

Summary: Returns the serial number of today.

Syntax: TODAY()

Semantics: This returns the current day's serial number, using current locale. This only returns the date, not the datetime value. For the specific time of day as well, use NOW 6.10.15.

6.10.20 WEEKDAY

Summary: Extracts the day of the week from a date; if text, uses current locale to convert to a date.

Syntax: WEEKDAY( DateParam D [ ; Integer Type = 1 ] )

Semantics: Returns the day of the week from a date D, as a number from 0 through 7. The exact meaning depends on the value of Type:

1.When Type is 1, Sunday is the first day of the week, with value 1; Saturday has value 7.
2.When Type is 2, Monday is the first day of the week, with value 1; Sunday has value 7.
3.When Type is 3, Monday is the first day of the week, with value 0; Sunday has value 6.
4.When Type is 11, Monday is the first day of the week, with value 1; Sunday has value 7.
5.When Type is 12, Tuesday is the first day of the week, with value 1; Monday has value 7.
6.When Type is 13, Wednesday is the first day of the week, with value 1; Tuesday has value 7.
7.When Type is 14, Thursday is the first day of the week, with value 1; Wednesday has value 7.
8.When Type is 15, Friday is the first day of the week, with value 1; Thursday has value 7.
9.When Type is 16, Saturday is the first day of the week, with value 1; Friday has value 7.
10. When Type is 17, Sunday is the first day of the week, with value 1; Saturday has value 7.

Table 6 - WEEKDAY

Weekday Type	1	2	3	11	12	13	14	15	16	17
Sunday	1	7	6	7	6	5	4	3	2	1
Monday	2	1	0	1	7	6	5	4	3	2
Tuesday	3	2	1	2	1	7	6	5	4	3
Wednesday	4	3	2	3	2	1	7	6	5	4
Thursday	5	4	3	4	3	2	1	7	6	5
Friday	6	5	4	5	4	3	2	1	7	6
Saturday	7	6	5	6	5	4	3	2	1	7

See also DAY 6.10.5, MONTH 6.10.13, YEAR 6.10.23

6.10.21 WEEKNUM

Summary: Determines the week number of the year for a given date.

Syntax: WEEKNUM( DateParam D [ ; Number Mode = 1 ] )

Constraints: 1 ≤ Mode ≤ 2, or 11 ≤ Mode ≤ 17, or Mode = 21, or Mode = 150

Semantics: Returns the number of the week in the year for the given date.

For Mode = {1, 2, 11, 12, ..., 17} the week containing January 1 is the first week of the year, and is numbered week 1. The week starts on {Sunday, Monday, Monday, Tuesday, ..., Sunday}.

Mode 21 and Mode 150 are both [ISO8601], the week starts on Monday and the week containing the first Thursday of the year is the first week of the year, and is numbered week 1.

See also DAY 6.10.5, MONTH 6.10.13, YEAR 6.10.23, WEEKDAY 6.10.20, ISOWEEKNUM 6.10.11

6.10.22 WORKDAY

Summary: Returns the date serial number which is a specified number of work days before or after an input date.

Syntax: WORKDAY( DateParam Date ; Number Offset [ ; [ DateSequence Holidays ] [ ; LogicalSequence Workdays ] ] )

Returns: DateTime

Semantics: Returns the date serial number for the day that is offset from the input Date parameter by the number of work days specified in the Offset parameter. If Offset is negative, the offset will return a date prior to Date. If Offset is positive, a date later Date is returned. If Offset is zero, then Date is returned.

Work days are defined as non-weekend, non-holiday days. By default, weekends are Saturdays and Sundays and there are no holidays.

The optional 4th parameter Workdays can be used to specify a different definition for the standard work week by passing in a list of numbers which define which days of the week are workdays (indicated by 0) or not (indicated by non-zero) in order Sunday, Monday,...,Saturday. If all seven numbers in Workdays are non-zero and Offset is also non-zero, WORKDAY returns an error.

Note: The default definition of the work week that excludes Saturday and Sunday and is: {1;0;0;0;0;0;1}. To define the workweek as excluding Friday and Saturday, the third parameter would be: {0;0;0;0;0;1;1}.

6.10.23 YEAR

Summary: Extracts the year from a date given in the current locale of the evaluator.

Syntax: YEAR( DateParam D )

Semantics: Parses a date-formatted string in the current locale's format and returns the year portion.

If a year is given as a two-digit number, as in "05-21-15", then the year returned is either 1915 or 2015, depending upon the break point in the calculation context. In an OpenDocument document, this break point is determined by HOST-NULL-YEAR.

Evaluators shall support extracting the year from a date beginning in 1900. Three-digit year numbers precede adoption of the Gregorian calendar, and may return either an Error or the year number. Four-digit year numbers preceding 1582 (inception of the Gregorian Calendar) may return either an Error or the year number. Four-digit year numbers following 1582 should return the year number.

See also MONTH 6.10.13, DAY 6.10.5, VALUE 6.13.34

6.10.24 YEARFRAC

Summary: Extracts the number of years (including fractional part) between two dates

Syntax: YEARFRAC( DateParam StartDate ; DateParam EndDate [ ; Basis B = 0 ] )

Semantics: Computes the fraction of the number of years between a StartDate and EndDate.

B indicates the day-count convention to use in the calculation. 4.11.7

6.11 External Access Functions

6.11.1 General

OpenFormula defines two functions, DDE and HYPERLINK, for accessing external data.

6.11.2 DDE

Summary: Returns data from a DDE request

Syntax: DDE( Text Server ; Text Topic ; Text Item [ ; Integer Mode = 0 ] )

Returns: Number|Text

Semantics: Performs a DDE request and returns its result. The request invokes the service Server on the topic named as Topic, requesting that it reply with the information on Item.

Evaluators may choose to not perform this function on every recalculation, but instead cache an answer and require a separate action to re-perform these requests. Evaluators shall perform this request on initial load when their security policies permit it.

Mode is an optional parameter that determines how the results are returned:

Table 7 - DDE

Mode	Effect
0 or missing	Data converted to number using VALUE in the number style's locale of the default table cell style
1	Data converted to number using VALUE in the English-US (en_US) locale
2	Data retrieved as text (not converted to number)

In an OpenDocument spreadsheet document the default table cell style is specified with table:default-cell-style-name. Its number:number-style specified by style:data-style-name specifies the locale to use in the conversion.

The DDE function is non-portable because it depends on availability of external programs (server parameter) and their interpretation of the topic and item parameters.

6.11.3 HYPERLINK

Summary: Creation of a hyperlink involving an evaluated expression.

Syntax: HYPERLINK( Text IRI [ ; Text|Number FunctionResult ] )

Returns: Text or Number

Semantics: The default for the second argument is the value of the first argument. The second argument value is returned.

In addition, hosting environments may interpret expressions containing HYPERLINK function calls as calling for an implementation-dependent creation of a hypertext link based on the expression containing the HYPERLINK function calls.

6.12 Financial Functions

6.12.1 General

The financial functions are defined for use in financial calculations.

An annuity is a recurring series of payments. A "simple annuity" is one where equal payments are made at equal intervals, and the compounding of interest occurs at those same intervals. The time between payments is called the "payment interval". Where payments are made at the end of the payment interval, it is called an "ordinary annuity". Where payments are made at the beginning of the payment interval, it is called an "annuity due". Periods are numbered starting at 1.

Financial functions defined in this standard use a cash flow sign convention where outgoing cash flows are negative and incoming cash flows are positive.

6.12.2 ACCRINT

Summary: Calculates the accrued interest for securities with periodic interest payments.

Syntax: ACCRINT( DateParam Issue ; DateParam First ; DateParam Settlement ; Number Coupon ; Number Par ; Integer Frequency [ ; Basis B = 0 [ ; Logical CalcMethod = TRUE ] ] )

Constraints: Issue < First < Settlement ; Coupon > 0; Par > 0

Semantics: Calculates the accrued interest for securities with periodic interest payments. ACCRINT supports short, standard, and long Coupon periods.

Table 8 - ACCRINT

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly
12	Monthly

If CalcMethod is TRUE (the default) then ACCRINT returns the sum of the accrued interest in each coupon period from issue date until settlement date. If CalcMethod is FALSE then ACCRINT returns the sum of the accrued interest in each coupon period from first interest date until settlement date. For each coupon period, the interest is Par * Coupon * YEARFRAC(start-of-period;end-of-period; B)

•Issue: The security's issue or dated date.
•First: The security's first interest date.
•Settlement: The security's settlement date.
•Coupon: The security's annual coupon rate.
•Par: The security's par value, that is, the principal to be paid at maturity.
•Frequency: The number of coupon payments per year.
•B: Indicates the day-count convention to use in the calculation. 4.11.7
•CalcMethod: A logical value that specifies how to treat the case where Settlement > First.

See also ACCRINTM 6.12.3, YEARFRAC 6.10.24

6.12.3 ACCRINTM

Summary: Calculates the accrued interest for securities that pay at maturity.

Syntax: ACCRINT( DateParam Issue ; DateParam Settlement ; Number Coupon ; Number Par [ ; Basis B = 0 ] )

Constraints: Coupon > 0; Par > 0

Semantics: Calculates the accrued interest for securities that pay at maturity.

•Issue: The security's issue or dated date.
•Settlement: The security's maturity date.
•Coupon: The security's annual coupon rate.
•Par: The security's par value, that is, the principal to be paid at maturity.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

6.12.4 AMORLINC

Summary: Calculates the amortization value for the French accounting system using linear depreciation (l'amortissement linéaire comptable) .

Syntax: AMORLINC( Number Cost ; DateParam PurchaseDate ; DateParam FirstPeriodEndDate ; Number Salvage ; Integer Period ; Number Rate [ ; Basis B = 0 ] )

Constraints: Cost > 0; PurchaseDate ≤ FirstPeriodEndDate; Salvage ≥ 0; Period ≥ 0; Rate > 0

Semantics: Calculates the amortization value for the French accounting system using linear depreciation.

•Cost: The value of the asset at the date of aquisition.
•PurchaseDate: The date of aquisition.
•FirstPeriodEndDate: The end date of the first depreciation period.
•Salvage: The value of the asset at the end of the depreciation life time.
•Period: Which period the depreciation should be calculated for.
•Rate: The rate of depreciation.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

When Period = 0:

AMORLINC = Cost \cdot Rate \cdot YEARFRAC (PurchaseDate, FirstPeriodEndDate, Basis)

For full periods, where Period > 0, the depreciation is Cost * Rate

t = \frac{Cost - Salvage}{Cost \cdot Rate}

For the last period, possibly a partial period: the depreciation = Cost - Salvage - accumulated-depreciation, where accumulated-depreciation is the sum of the depreciation in period 0 plus any full period depreciations.

AMORLINC = Cost \cdot Rate

When Period > depreciated life of the asset, i.e., when Period > (Cost - Salvage) / (Cost * Rate) then the depreciation is 0.

AMORLINC = 0

Note: The behavior of this function is implementation-defined in cases where PurchaseDate = FirstPeriodEndDate.

See also DB 6.12.13, DDB 6.12.14, YEARFRAC 6.10.24

6.12.5 COUPDAYBS

Summary: Calculates the number of days between the beginning of the coupon period that contains the settlement date and the settlement date.

Syntax: COUPDAYBS( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Semantics: Calculate the number of days from the beginning of the coupon period to the settlement date.

Table 9 - COUPDAYBS

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

•Settlement: The settlement date.
•Maturity: The maturity date.
•Frequency: The number of coupon payments per year.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

See also COUPDAYS 6.12.6 , COUPDAYSNC 6.12.7 , COUPNCD 6.12.7 , COUPNUM 6.12.9 , COUPPCD 6.12.10

6.12.6 COUPDAYS

Summary: Calculates the number of days in a coupon period that contains the settlement date.

Syntax: COUPDAYS( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Semantics: Calculates the number of days in the coupon period containing the settlement date.

Table 10 - COUPDAYS

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

•Settlement: The settlement date.
•Maturity: The maturity date.
•Frequency: The number of coupon payments per year.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

See also COUPDAYBS 6.12.5 , COUPDAYSNC 6.12.7 , COUPNCD 6.12.7 , COUPNUM 6.12.9 , COUPPCD 6.12.10

6.12.7 COUPDAYSNC

Summary: Calculates the number of days between a settlement date and the next coupon date.

Syntax: COUPDAYNC( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Semantics: Calculates the number of days between the settlement date and the next coupon date.

Table 11 - COUPDAYSNC

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

•Settlement: The settlement date.
•Maturity: The maturity date.
•Frequency: The number of coupon payments per year.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

See also COUPDAYBS 6.12.5 , COUPDAYS 6.12.6 , COUPNCD 6.12.7 , COUPNUM 6.12.9 , COUPPCD 6.12.10

6.12.8 COUPNCD

Summary: Calculates the next coupon date following a settlement.

Syntax: COUPNCD( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Frequency is the number of coupon payments per year. Frequency is one of the following values:

Table 12 - COUPNCD

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

Semantics: Calculates the next coupon date after the Settlement date based on the Maturity (expiration) date of the asset, the Frequency of coupon payments and the day-count B.

B indicates the day-count convention to use in the calculation. 4.11.7

6.12.9 COUPNUM

Summary: Calculates the number of outstanding coupons between settlement and maturity dates.

Syntax: COUPNUM( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Constraints: Frequency is the number of coupon payments per year. Frequency is one of the following values:

Table 13 - COUPNUM

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

Semantics: Calculates the number of coupons in the interval between the Settlement and the Maturity (expiration) date of the asset, the Frequency of coupon payments and the day-count B.

B indicates the day-count convention to use in the calculation. 4.11.7

See also COUPDAYBS 6.12.5, COUPDAYS 6.12.6, COUPDAYSNC 6.12.7, COUPNCD 6.12.7, COUPPCD 6.12.10

6.12.10 COUPPCD

Summary: Calculates the next coupon date prior a settlement.

Syntax: COUPPCD( DateParam Settlement ; DateParam Maturity ; Integer Frequency [ ; Basis B = 0 ] )

Frequency is the number of coupon payments per year. Frequency is one of the following values:

Table 14 - COUPPCD

Frequency	Frequency of coupon payments
1	Annual
2	Semiannual
4	Quarterly

Semantics: Calculates the next coupon date prior to the Settlement date based on the Maturity (expiration) date of the asset, the Frequency of coupon payments and the day-count B.

B indicates the day-count convention to use in the calculation. 4.11.7

See also COUPDAYBS 6.12.5, COUPDAYS 6.12.6, COUPDAYSNC 6.12.7, COUPNCD 6.12.7, COUPNUM 6.12.9

6.12.11 CUMIPMT

Summary: Calculates a cumulative interest payment.

Syntax: CUMIPMT( Number Rate ; Number Periods ; Number Value ; Integer Start ; Integer End ; Integer Type )

Constraints: Rate > 0; Value > 0; 1 ≤ Start ≤ End ≤ Periods

Type is one of the following values:

Table 15 - CUMIPMT

Type	Maturity date
0	due at the end
1	due at the beginning

Semantics: Calculates the cumulative interest payment.

•Rate: The interest rate per period.
•Periods: The number of periods.
•Value: The current value of the loan.
•Start: The starting period.
•End: The end period.
•Type: The maturity date, the beginning or the end of a period.

CUMIPMT = \sum_{p = Start}^{End} IPMT (Rate, p, Periods, Value, 0, Type)

6.12.12 CUMPRINC

Summary: Calculates a cumulative principal payment.

Syntax: CUMPRINC( Number Rate ; Number Periods ; Number Value ; Integer Start ; Integer End ; Integer Type )

Constraints: Type is one of the following values:

Table 16 - CUMPRINC

Type	Maturity date
0	due at the end
1	due at the beginning

Type	Maturity date
0	due at the end
1	due at the beginning

Semantics: Calculates the cumulative principal payment.

•Rate: The interest rate per period.
•Periods: The number of periods.
•Value: The current value of the loan.
•Start: The starting period.
•End: The end period.
•Type: The maturity date, the beginning or the end of a period.

CUMPRINC = \sum_{p = Start}^{End} PPMT (Rate, p, Periods, Value, 0, Type)

6.12.13 DB

Summary: Compute the depreciation allowance of an asset.

Syntax: DB( Number Cost ; Number Salvage ; Integer LifeTime ; Number Period [ ; Number Month = 12 ] )

Constraints: Cost > 0, Salvage ≥ 0, LifeTime > 0; Period > 0; 0 < Month < 13

Semantics: Calculate the depreciation allowance of an asset with an initial value of Cost, an expected useful LifeTime, and a final Salvage value at a specified Period of time, using the fixed-declining balance method. The parameters are:

•Cost: the total amount paid for the asset.
•Salvage: the salvage value at the end of the LifeTime.
•LifeTime: the number of periods that the depreciation will occur over. A positive integer.
•Period: the time period for which you want to find the depreciation allowance, in the same units as LifeTime.
•Month: (optional) the number of months in the first year of depreciation, assumed to be 12, if not specified. If a value is specified for Month, LifeTime and Period are assumed to be measured in years.

The rate is calculated as follows:

rate = 1 - {(\frac{Salvage}{Cost})}^{\frac{1}{LifeTime}}

and is rounded to 3 decimals.

For the first period the residual value is

{value}_{1} = Cost (1 - \frac{Month}{12} \cdot rate)

For all periods, where Period ≤ LifeTime, the residual value is calculated by

{value}_{Period} = {value}_{Period - 1} \cdot (1 - rate)

If Month was specified, the residual value for the period after LifeTime becomes

{value}_{LifeTime + 1} = {value}_{LifeTime} \cdot (1 - (1 - \frac{Month}{12}) \cdot rate)

The depreciation allowance for the first period is

{DB}_{1} = Cost - {value}_{1}

For all other periods the allowance is calculated by

{DB}_{period} = {value}_{period} - {value}_{period - 1}

For all periods, where Period > LifeTime + 1 – INT(Month / 12), the depreciation allowance is zero.

6.12.14 DDB

Summary: Compute the amount of depreciation at a given period of time.

Syntax: DDB( Number Cost ; Number Salvage ; Number LifeTime ; Number Period [ ; Number DeclinationFactor = 2 ] )

Constraints: Cost ≥ 0, Salvage ≥ 0, Salvage ≤ Cost, 1 ≤ Period ≤ LifeTime, DeclinationFactor > 0

Semantics: Compute the amount of depreciation of an asset at a given period of time. The parameters are:

•Cost: the total amount paid for the asset.
•Salvage: the salvage value at the end of the LifeTime
•LifeTime: the number of periods that the depreciation will occur over.
•Period: the period for which a depreciation value is specified.
•DeclinationFactor: the method of calculating depreciation, the rate at which the balance declines. Defaults to 2. If 2, double-declining balance is used.

To calculate depreciation, DDB uses a fixed rate. When DeclinationFactor = 2 this is the double-declining-balance method (because it is double the straight-line rate that would depreciate the asset to zero). The rate is given by:

rate = \frac{DeclinationFactor}{LifeTime}

The depreciation each period is calculated as

depreciation_of_period = MIN( book_value_at_start_of_ period * rate; book_value_at_start_of_ period - Salvage )

Thus the asset depreciates at rate until the book value is Salvage value.

{BookValueAtStartOfPeriod}_{Period} = Cost - \sum_{i = 1}^{Period - 1} {DepreciationOfPeriod}_{i}

To allow also non-integer Period values this algorithm may be used:

\begin{matrix} rate = \frac{DeclinationFactor}{LifeTime} \\ if rate \geq 1 then \\ { \\ rate = 1 \\ if Period = 1 then \\ oldValue = Cost \\ else \\ oldValue = 0 \\ endif \\ } \\ else \\ { \\ oldValue = Cost \cdot {(1 - rate)}^{Period - 1} \\ } \\ endif \\ newValue = Cost \cdot {(1 - rate)}^{Period} \\ if newValue < Salvage then \\ DDB = oldValue - Salvage \\ else \\ DDB = oldValue - newValue \\ endif \\ if DDB < 0 then \\ DDB = 0 \\ endif \end{matrix}

If Period is an Integer number, the relation between DDB and VDB is:
DDB( Cost ; Salvage ; LifeTime ; Period ; DeclinationFactor )
equals
VDB( Cost ; Salvage ; LifeTime ; Period - 1 ; Period ; DeclinationFactor ; TRUE )

6.12.15 DISC

Summary: Returns the discount rate of a security.

Syntax: DISC( DateParam Settlement ; DateParam Maturity ; Number Price ; Number Redemption [ ; Basis B = 0 ] )

Semantics: Calculates the discount rate of a security.

•Settlement: The settlement date of the security.
•Maturity: The maturity date.
•Price: The price of the security.
•Redemption: The redemption value of the security.
•B: Indicates the day-count convention to use in the calculation. 4.11.7

DISC = \frac{\frac{Redemption - Price}{Redemption}}{YEARFRAC (Settlement, Maturity, B)}

6.12.16 DOLLARDE

Summary: Converts a fractional dollar representation into a decimal representation.

Syntax: DOLLARDE( Number Fractional ; Integer Denominator )

Constraints: Denominator > 0

Semantics: Converts a fractional dollar representation into a decimal representation.

•Fractional: Decimal fraction.
•Denominator: The denominator of the fraction.

DOLLARDE = TRUNC (Fractional) + \frac{Fractional - TRUNC (Fractional)}{Denominator}

6.12.17 DOLLARFR

Summary: Converts a decimal dollar representation into a fractional representation.

Syntax: DOLLARFR( Number Decimal ; Integer Denominator )

Constraints: Denominator > 0

Semantics: Converts a decimal dollar representation into a fractional representation.

•Decimal: A decimal number.
•Denominator: The denominator of the fraction.

DOLLARFR = TRUNC (Decimal) + (Decimal - TRUNC (Decimal)) \cdot Denominator

6.12.18 DURATION

Summary: Returns the Macaulay duration of a fixed interest security in years

Syntax: DURATION( Date Settlement ; Date Maturity ; Number Coupon ; Number Yield ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Yield ≥0, Coupon ≥ 0, Settlement ≤ Maturity; Frequency = 1, 2, 4

Semantics: Computes the Macaulay duration, given:

•Settlement: the date of purchase of the security
•Maturity: the date when the security matures
•Coupon: the annual nominal rate of interest
•Yield: the annual yield of the security
•Frequency: number of interest payments per year
•B: Indicates the day-count convention to use in the calculation. 4.11.7

6.12.19 EFFECT

Summary: Returns the net annual interest rate for a nominal interest rate.

Syntax: EFFECT( Number Rate ; Integer Payments )

Constraints: Rate ≥ 0; Payments > 0

Semantics: Nominal interest refers to the amount of interest due at the end of a calculation period. Effective interest increases with the number of payments made. In other words, interest is often paid in installments (for example, monthly or quarterly) before the end of the calculation period.

•Rate: The interest rate per period.
•Payments: The number of payments per period.

EFFECT = {(1 + \frac{Rate}{Payments})}^{Payments} - 1

6.12.20 FV

Summary: Compute the future value (FV) of an investment.

Syntax: FV( Number Rate ; Number Nper ; Number Payment [ ; [ Number Pv = 0 ] [ ; Number PayType = 0 ] ] )

Semantics: Computes the future value of an investment. The parameters are:

•Rate: the interest rate per period.
•Nper: the total number of payment periods.
•Payment: the payment made in each period.
•Pv: the present value; default is 0.
•PayType: the type of payment, defaults to 0. It is 0 if payments are due at the end of the period; 1 if they are due at the beginning of the period.

See PV 6.12.41 for the equation this solves.

See also PV 6.12.41, NPER 6.12.29, PMT 6.12.36, RATE 6.12.42

6.12.21 FVSCHEDULE

Summary: Returns the accumulated value given starting capital and a series of interest rates.

Syntax: FVSCHEDULE( Number Principal ; NumberSequence Schedule )

Semantics: Returns the accumulated value given starting capital and a series of interest rates, as follows:

Principle \cdot \prod_{i = 1}^{N} (1 + Schedule [i])

See also PV 6.12.41, NPER 6.12.29, PMT 6.12.36, RATE 6.12.42

6.12.22 INTRATE

Summary: Computes the interest rate of a fully vested security.

Syntax: INTRATE( Date Settlement ; Date Maturity ; Number Investment ; Number Redemption [ ; Basis Basis = 0 ] )

Semantics: Calculates the annual interest rate that results when an item is purchased at the investment price and sold at the redemption price. No interest is paid on the investment. The parameters are:

•Settlement: the date of purchase of the security.
•Maturity: the date on which the security is sold.
•Investment: the purchase price.
•Redemption: the selling price.
•Basis: indicates the day-count convention to use in the calculation. 4.11.7

The return value for this function is:

INTRATE = \frac{\frac{Redemption - Investment}{Investment}}{YEARFRAC (Settlement; Maturity; B)}

See also RECEIVED 6.12.43, YEARFRAC 6.10.24

6.12.23 IPMT

Summary: Returns the amount of an annuity payment going towards interest.

Syntax: IPMT( Number Rate ; Number Period ; Number Nper ; Number PV [ ; Number FV = 0 [ ; Number Type = 0 ] ] )

Semantics: Computes the interest portion of an amortized payment for a constant interest rate and regular payments. The interest payment is the interest rate multiplied by the balance at the beginning of the period. The parameters are:

•Rate: The periodic interest rate.
•Period: The period for which the interest payment is computed.
•Nper: The total number of periods for which the payments are made
•PV: The present value (e.g. The initial loan amount).
•FV: The future value (optional) at the end of the periods. Zero if omitted.
•Type: the due date for the payments (optional). Zero if omitted. If Type is 1, then payments are made at the beginning of each period. If Type is 0, then payments are made at the end of each period.

6.12.24 IRR

Summary: Compute the internal rate of return for a series of cash flows.

Syntax: IRR( NumberSequence Values [ ; Number Guess = 0.1 ] )

Semantics: Compute the internal rate of return for a series of cash flows.

If provided, Guess is an estimate of the interest rate to start the iterative computation. If omitted, the value 0.1 (10%) is assumed.

The result of IRR is the rate at which the NPV() function will return zero with the given values.

There is no closed form for IRR. Evaluators may return an approximate solution using an iterative method, in which case the Guess parameter may be used to initialize the iteration. If the evaluator is unable to converge on a solution given a particular Guess, it may return an Error.

6.12.25 ISPMT

Summary: Compute the interest payment of an amortized loan for a given period.

Syntax: ISPMT( Number Rate ; Number Period ; Number Nper ; Number Pv )

Semantics: Computes the interest payment of an amortized loan for a given period. The parameters are:

•Rate: the interest rate per period.
•Period: the period for which the interest is computed
•Nper: the total number of payment periods.
•Pv: the amount of the investment

See also PV 6.12.41, FV 6.12.20, NPER 6.12.29, PMT 6.12.36, RATE 6.12.42

6.12.26 MDURATION

Summary: Returns the modified Macaulay duration of a fixed interest security in years.

Syntax: MDURATION( Date Settlement ; Date Maturity ; Number Coupon ; Number Yield ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Yield ≥ 0, Coupon ≥ 0, Settlement ≤ Maturity; Frequency = 1, 2, 4

Semantics: Computes the modified Macaulay duration, given:

•Settlement: the date of purchase of the security
•Maturity: the date when the security matures
•Coupon: the annual nominal rate of interest
•Yield: the annual yield of the security
•Frequency: number of interest payments per year
•B: Indicates the day-count convention to use in the calculation. 4.11.7

The modified duration is computed as follows:

\begin{matrix} duration = DURATION (Settlement, Maturity, Coupon, Yield, Frequency, B) \\ MDURATION = \frac{duration}{1 + (\frac{Yield}{Frequency})} \end{matrix}

6.12.27 MIRR

Summary: Returns the modified internal rate of return (IRR) of a series of periodic investments.

Syntax: MIRR( Array Values ; Number Investment ; Number ReinvestRate )

Constraints: Values shall contain at least one positive value and at least one negative value.

Semantics: Values is a series of periodic income (positive values) and payments (negative values) at regular intervals (Text and Empty cells are ignored). Investment is the rate of interest of the payments (negative values); ReinvestRate is the rate of interest of the reinvestment (positive values).

Computes the modified internal rate of return, which is:

{(\frac{- NPV (ReinvestRate, Values > 0) * {(1 + ReinvestRate)}^{n}}{NPV (Investment; Values < 0) * (1 + Investment)})}^{(\frac{1}{n - 1})} - 1

where N is the number of incomes and payments in Values (total).

6.12.28 NOMINAL

Summary: Compute the annual nominal interest rate.

Syntax: NOMINAL( Number EffectiveRate ; Integer CompoundingPeriods )

Constraints: EffectiveRate > 0 , CompoundingPeriods > 0

Semantics: Returns the annual nominal interest rate based on the effective rate and the number of compounding periods in one year. The parameters are:

•EffectiveRate: effective rate
•CompoundingPeriods: the compounding periods per year

Suppose that P is the present value, m is the compounding periods per year, the future value after one year is

P * {(1 + \frac{NOMINAL}{m})}^{m}

The mapping between nominal rate and effective rate is

EFFECT = {(1 + \frac{NOMINAL}{m})}^{m} - 1

6.12.29 NPER

Summary: Compute the number of payment periods for an investment.

Syntax: NPER( Number Rate ; Number Payment ; Number Pv [ ; [ Number Fv ] [ ; Number PayType ] ] )

Semantics: Computes the number of payment periods for an investment. The parameters are:

•Rate: the constant interest rate.
•Payment: the payment made in each period.
•Pv: the present value of the investment.
•Fv: the future value; default is 0.
•PayType: the type of payment, defaults to 0. It is 0 if payments are due at the end of the period; 1 if they are due at the beginning of the period.

If Rate is 0, then NPER solves this equation:

Pv = - Fv - (Payment * NPER)

If Rate is non-zero, then NPER solves this equation:

0 = Pv \cdot {(1 + Rate)}^{NPER} + \frac{Payment \cdot (1 + Rate \cdot PaymentType) \cdot ({(1 + Rate)}^{NPER} - 1)}{Rate} + Fv

Evaluators claiming to support the “Medium” or “Large” set shall support negative rates; evaluators only claiming to support the “Small” set need not.

See also FV 6.12.20, RATE 6.12.42, PMT 6.12.36, PV 6.12.41

6.12.30 NPV

Summary: Compute the net present value (NPV) for a series of periodic cash flows.

Syntax: NPV( Number Rate ; { NumberSequenceList Values }+ )

Semantics: Computes the net present value for a series of periodic cash flows with the discount rate Rate. Values should be positive if they are received as income, and negative if the amounts are paid as outgo. Because the result is affected by the order of values, evaluators shall evaluate arguments in the order given and range reference and array arguments row-wise starting from top left.

If N is the number of values in Values, the formula for NPV is:

NPV = \sum_{i = 1}^{N} \frac{{Values}_{i}}{{(1 + Rate)}^{i}}

See also FV 6.12.20, IRR 6.12.24, NPER 6.12.29, PMT 6.12.36, PV 6.12.41, XNPV 6.12.52

6.12.31 ODDFPRICE

Summary: Compute the value of a security per 100 currency units of face value. The security has an irregular first interest date.

Syntax: ODDFPRICE( DateParam Settlement ; DateParam Maturity ; DateParam Issue ; DateParam First ; Number Rate ; Number Yield ; Number Redemption ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Rate, Yield, and Redemption should be greater than 0.

Summary: Compute the yield of a security per 100 currency units of face value. The security has an irregular first interest date.

•Settlement: the settlement/purchase date of the security
•Maturity: the maturity/expiry date of the security
•Issue: the issue date of the security
•First: the first coupon date of the security
•Rate: the interest rate of the security
•Yield: the annual yield of the security
•Redemption: the redemption value per 100 currency units face value
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also ODDLPRICE 6.12.33 , ODDFYIELD 6.12.32

6.12.32 ODDFYIELD

Syntax: ODDFYIELD( DateParam Settlement ; DateParam Maturity ; DateParam Issue ; DateParam First ; Number Rate ; Number Price ; Number Redemption ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Rate, Price, and Redemption should be greater than 0. Maturity > First > Settlement > Issue.

Summary: Compute the value of a security per 100 currency units of face value. The security has an irregular last interest date.

•Settlement: the settlement/purchase date of the security
•Maturity: the maturity/expiry date of the security
•Issue: the issue date of the security
•First: the first coupon date of the security
•Rate: the interest rate of the security
•Price: the price of the security
•Redemption: the redemption value per 100 currency units face value
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also ODDLYIELD 6.12.34 , ODDFPRICE 6.12.31

6.12.33 ODDLPRICE

Syntax: ODDLPRICE( DateParam Settlement ; DateParam Maturity ; DateParam Last ; Number Rate ; Number AnnualYield ; Number Redemption ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Rate, AnnualYield, and Redemption should be greater than 0. The Maturity date should be greater than the Settlement date, and the Settlement should be greater than the last interest date.

Summary: Compute the yield of a security which has an irregular last interest date.

•Settlement: the settlement/purchase date of the security
•Maturity: the maturity/expiry date of the security
•Last: the last interest date of the security
•Rate: the interest rate of the security
•AnnualYield: the annual yield of the security
•Redemption: the redemption value per 100 currency units face value
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly
•B: indicates the day-count convention to use in the calculation. 4.11.7

6.12.34 ODDLYIELD

Syntax: ODDLYIELD( DateParam Settlement ; DateParam Maturity ; DateParam Last ; Number Rate ; Number Price ; Number Redemption ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Rate, Price, and Redemption should be greater than 0.

Summary: Returns the number of periods required by an investment to realize a specified value.

•Settlement: the settlement/purchase date of the security
•Maturity: the maturity/expiry date of the security
•Last: the last interest date of the security
•Rate: the interest rate of the security
•Price: the price of the security
•Redemption: the redemption value per 100 currency units face value
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also ODDLPRICE 6.12.33 , ODDFYIELD 6.12.32

6.12.35 PDURATION

Syntax: PDURATION( Number Rate ; Number CurrentValue ; Number SpecifiedValue )

Constraints: Rate > 0; CurrentValue > 0; SpecifiedValue > 0

Semantics: Calculates the number of periods for attaining a certain value SpecifiedValue, starting from CurrentValue and using the interest rate Rate.

•Rate: The interest rate per period.
•CurrentValue: The current value of the investment.
•SpecifiedValue: The value, that should be reached.

PDURATION = \frac{\log (SpecifiedValue) - \log (CurrentValue)}{\log (Rate + 1)}

6.12.36 PMT

Summary: Compute the payment made each period for an investment.

Syntax: PMT( Number Rate ; Integer Nper ; Number Pv [ ; [ Number Fv = 0 ] [ ; Number PayType = 0 ] ] )

Semantics: Computes the payment made each period for an investment. The parameters are:

Constraints: Nper > 0

•Rate: the interest rate per period.
•Nper: the total number of payment periods.
•Pv: the present value of the investment.
•Fv: the future value of the investment; default is 0.
•PayType: the type of payment, defaults to 0. It is 0 if payments are due at the end of the period; 1 if they are due at the beginning of the period. With PayType = 1 the first payment is made on the same day the loan is taken out.

If Rate is 0, the following equation is solved:

Pv = - Fv - (PMT * Nper)

If Rate is nonzero, then PMT solves this equation:

0 = Pv \cdot {(1 + Rate)}^{Nper} + \frac{PMT \cdot (1 + Rate \cdot PayType) \cdot ({(1 + Rate)}^{Nper} - 1)}{Rate} + Fv

See also FV 6.12.20, NPER 6.12.29, PV 6.12.41, RATE 6.12.42

6.12.37 PPMT

Summary: Calculate the payment for a given period on the principal for an investment at a given interest rate and constant payments.

Syntax: PPMT( Number Rate ; Integer Period ; Integer Nper ; Number Present [ ; Number Future = 0 [ ; Number Type = 0 ] ] )

Constraints: Rate and Present should be greater than 0. 0 < Period < Nper.

Summary: Calculates a quoted price for an interest paying security, per 100 currency units of face value.

•Rate: the interest rate.
•Period: the given period that the payment returned is for.
•Nper: the total number of periods.
•Present: the present value.
•Future: optional, the future value specified after Nper periods. The default value is 0.
•Type: optional, 0 or 1, respectively for payment at the end or at the beginning of a period. The default value is 0.

6.12.38 PRICE

Syntax: PRICE( DateParam Settlement ; DateParam Maturity ; Number Rate ; Number AnnualYield ; Number Redemption ; Number Frequency [ ; Basis Bas = 0 ] )

Constraints: Rate, AnnualYield, and Redemption should be greater than 0; Frequency = 1, 2 or 4.

Semantics: If A is the number of days from the Settlement date to next coupon date, B is the number of days of the coupon period that the Settlement is in, C is the number of coupons between Settlement date and Redemption date, D is the number of days from beginning of coupon period to Settlement date, then PRICE is calculated as

PRICE = \frac{Redemption}{{(1 + \frac{Yield}{Frequency})}^{C - 1 + \frac{A}{B}}} + \sum_{k = 1}^{C} \frac{\frac{100 * Rate}{Frequency}}{{(1 + \frac{Yield}{Frequency})}^{k - 1 + \frac{A}{B}}} - 100 * \frac{Rate}{Frequency} * \frac{D}{B}

The parameters are:

•Settlement: the settlement/purchase date of the security.
•Maturity: the maturity/expiry date of the security.
•Rate: the interest rate of the security.
•AnnualYield: a measure of the annual yield of a security (compounded at each interest payment).
•Redemption: the redemption value per 100 currency units face value.
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly.
•Bas: indicates the day-count convention to use in the calculation. 4.11.7

See also PRICEDISC 6.12.39, PRICEMAT 6.12.40

6.12.39 PRICEDISC

Summary: Calculate the price of a security with a discount per 100 currency units of face value.

Syntax: PRICEDISC( DateParam Settlement ; DateParam Maturity ; Number Discount ; Number Redemption [ ; Basis B = 0 ] )

Constraints: Discount and Redemption should be greater than 0.

Summary: Calculate the price per 100 currency units of face value of the security that pays interest on the maturity date.

•Settlement: the settlement/purchase date of the security.
•Maturity: the maturity/expiry date of the security.
•Discount: the discount rate of the security.
•Redemption: the redemption value per 100 currency units face value.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also PRICE 6.12.38, PRICEMAT 6.12.40, YIELDDISC 6.12.54

6.12.40 PRICEMAT

Syntax: PRICEMAT( DateParam Settlement ; DateParam Maturity ; DateParam Issue ; Number Rate ; Number AnnualYield [ ; Basis B = 0 ] )

Constraints: Settlement < Maturity, Rate ≥ 0, AnnualYield ≥ 0

Summary: Compute the present value (PV) of an investment.

•Settlement: the settlement/purchase date of the security.
•Maturity: the maturity/expiry date of the security.
•Issue: the issue date of the security.
•Rate: the interest rate of the security.
•AnnualYield: the annual yield of the security.
•B: indicates the day-count convention to use in the calculation. 4.11.7

If both, Rate and AnnualYield, are 0, the return value is 100.

See also PRICEDISC 6.12.39, PRICEMAT 6.12.40

6.12.41 PV

Syntax: PV( Number Rate ; Number Nper ; Number Payment [ ; [ Number Fv = 0 ] [ ; Number PayType = 0 ] ] )

Semantics: Computes the present value of an investment. The parameters are:

•Rate: the interest rate per period.
•Nper: the total number of payment periods.
•Payment: the payment made in each period.
•Fv: the future value; default is 0.
•PayType: the type of payment, defaults to 0. It is 0 if payments are due at the end of the period; 1 if they are due at the beginning of the period.

If Rate is 0, then:

PV = - Fv - (Payment * Nper)

If Rate is nonzero, then PV solves this equation:

0 = PV \cdot {(1 + Rate)}^{Nper} + \frac{Payment \cdot (1 + Rate \cdot PayType) \cdot ({(1 + Rate)}^{Nper} - 1)}{Rate} + Fv

See also FV 6.12.20, NPER 6.12.29, PMT 6.12.36, RATE 6.12.42

6.12.42 RATE

Summary: Compute the interest rate per period of an investment.

Syntax: RATE( Number Nper ; Number Payment ; Number Pv [ ; [ Number Fv = 0 ] [ ; [ Number PayType = 0 ] [ ; Number Guess = 0.1 ] ] ] )

Constraints: If Nper is 0 or less than 0, the result is an Error.

Semantics: Computes the interest rate of an investment. The parameters are:

•Nper: the total number of payment periods.
•Payment: the payment made in each period.
•Pv: the present value of the investment.
•Fv: the future value; default is 0.
•PayType: the type of payment, defaults to 0. It is 0 if payments are due at the end of the period; 1 if they are due at the beginning of the period.
•Guess: An estimate of the interest rate to start the iterative computation. If omitted, 0.1 (10%) is assumed.

RATE solves this equation:

\begin{matrix} 0 = Fv + Pv \cdot {(1 + Rate)}^{Nper} \\ + \frac{Payment \cdot (1 + Rate \cdot PayType) \cdot ({(1 + Rate)}^{Nper} - 1)}{Rate} \end{matrix}

See also FV 6.12.20, NPER 6.12.29, PMT 6.12.36, PV 6.12.41

6.12.43 RECEIVED

Summary: Calculates the amount received at maturity for a zero coupon bond.

Syntax: RECEIVED( DateParam Settlement ; DateParam Maturity ; Number Investment ; Number Discount [ ; Basis B = 0 ] )

Constraints: Investment and Discount should be greater than 0, Settlement < Maturity

Settlement: the settlement/purchase date of the security

•Maturity: the maturity/expiry date of the security
•Investment: the amount of investment in the security
•Discount: the discount rate of the security
•B: indicates the day-count convention to use in the calculation. 4.11.7

The returned value is:

RECEIVED = \frac{Investment}{1 - Discount \cdot YEARFRAC (Settlement; Maturity; B)}

6.12.44 RRI

Summary: Returns an equivalent interest rate when an investment increases in value.

Syntax: RRI( Number Nper ; Number Pv ; Number Fv )

Semantics: Returns the interest rate given Nper (the number of periods), Pv (present value), and Fv (future value), calculated as follows:

Constraints: Nper > 0

{(\frac{Fv}{Pv})}^{(1 / Nper)} - 1

See also FV 6.12.20, NPER 6.12.29, PMT 6.12.36, PV 6.12.41, RATE 6.12.42

6.12.45 SLN

Summary: Compute the amount of depreciation at a given period of time using the straight-line depreciation method.

Syntax: DDB( Number Cost ; Number Salvage ; Number LifeTime )

Semantics: Compute the amount of depreciation of an asset at a given period of time using straight-line depreciation. The parameters are:

•Cost: the total amount paid for the asset.
•Salvage: the salvage value at the end of the LifeTime (often 0)
•LifeTime: the number of periods that the depreciation will occur over. A positive integer.

For alternative methods to compute depreciation, see DDB 6.12.14.

6.12.46 SYD

Summary: Compute the amount of depreciation at a given period of time using the Sum-of-the-Years'-Digits method.

Syntax: SYD( Number Cost ; Number Salvage ; Number LifeTime ; Number Period )

Semantics: Compute the amount of depreciation of an asset at a given period of time using the Sum-of-the-Years'-Digits method. The parameters are:

•Cost: the total amount paid for the asset.
•Salvage: the salvage value at the end of the LifeTime (often 0).
•LifeTime: the number of periods that the depreciation will occur over. A positive integer.
•Period: the period for which the depreciation value is specified.

SYD = \frac{(Cost - Salvage) \cdot (LifeTime + 1 - Period) \cdot 2}{(LifeTime + 1) \cdot LifeTime}

For other methods of computing depreciation, see DDB 6.12.14.

6.12.47 TBILLEQ

Summary: Compute the bond-equivalent yield for a treasury bill.

Syntax: TBILLEQ( DateParam Settlement ; DateParam Maturity ; Number Discount )

Constraints: The maturity date should be less than one year beyond settlement date. Discount is any positive value.

Semantics: The parameters are defined as:

•Settlement: the settlement/purchase date of the treasury bill.
•Maturity: the maturity/expiry date of the treasury bill.
•Discount: the discount rate of the treasury bill.

TBILLEQ is calculated as

TBILLEQ = \frac{365 \cdot rate}{360 - (rate \cdot DSM)}

where DSM is the number of days between settlement and maturity computed according to the 360 days per year basis (basis 2, 4.11.7)

See also TBILLPRICE 6.12.48, TBILLYIELD 6.12.49

6.12.48 TBILLPRICE

Summary: Compute the price per 100 face value for a treasury bill.

Syntax: TBILLPRICE( DateParam Settlement ; DateParam Maturity ; Number Discount )

Constraints: The maturity date should be less than one year beyond settlement. Discount is any positive value.

Summary: Compute the yield for a treasury bill.

•Settlement: the settlement/purchase date of the treasury bill.
•Maturity: the maturity/expiry date of the treasury bill.
•Discount: the discount rate of the treasury bill.

See also TBILLEQ 6.12.47, TBILLYIELD 6.12.49

6.12.49 TBILLYIELD

Syntax: TBILLYIELD( DateParam Settlement ; DateParam Maturity ; Number Price )

Constraints: The maturity date should be less than one year beyond settlement. Price is any positive value.

Summary: Calculates the depreciation allowance of an asset with an initial value, an expected useful life, and a final value of salvage for a period specified, using the variable-rate declining balance method..

•Settlement: the settlement/purchase date of the treasury bill.
•Maturity: the maturity/expiry date of the treasury bill.
•Price: the price of the treasury bill per 100 face value

See also TBILLEQ 6.12.47, TBILLPRICE 6.12.48

6.12.50 VDB

Syntax: VDB( Number Cost ; Number Salvage ; Number LifeTime ; Number StartPeriod ; Number EndPeriod [ ; Number DepreciationFactor = 2 [ ; Logical NoSwitch = FALSE ] ] )

Constraints: Salvage < Cost, LifeTime > 0, 0 ≤ StartPeriod ≤ LifeTime, StartPeriod ≤ EndPeriod ≤ LifeTime, DepreciationFactor ≥ 0

VDB allows for the use of an initialPeriod option to calculate depreciation for the period the asset is placed in service. VDB uses the fractional part of StartPeriod and EndPeriod to determine the initialPeriod option. If both StartPeriod and EndPeriod have fractional parts, then VDB uses the fractional part of StartPeriod.

•Cost is the amount paid for the asset. Cost can be any value greater than Salvage.
•Salvage is the value of the asset at the end of its life. Salvage can be any value.
•LifeTime is the number of periods the asset takes to depreciate to its salvage value. LifeTime can be any value greater than 0.
•StartPeriod is the point in the asset's life when you want to begin calculating depreciation. StartPeriod can be any value greater than or equal to 0, but cannot be greater than LifeTime.
•EndPeriod is the point in the asset's life when you want to stop calculating depreciation. EndPeriod can be any value greater than StartPeriod.
•StartPeriod and EndPeriod correspond to the asset's life, relative to the fiscal period. For example, if you want to find the first year's depreciation of an asset purchased at the beginning of the second quarter of a fiscal year, StartPeriod would be 0 and EndPeriod would be 0.75 (1 minus 0.25 of a year).

DepreciationFactor is an optional argument that specifies the percentage of straight-line depreciation you want to use as the depreciation rate. If you omit this argument, VDB uses 2, which is the double-declining balance rate. DepreciationFactor can be any value greater than or equal to 0; commonly used rates are 1.25, 1.50, 1.75, and 2.

NoSwitch is an optional argument that you include if you do not want VDB to switch to straight-line depreciation for the remaining useful life. Normally, declining-balance switches to such a straight-line calculation when it is greater than the declining-balance calculation.

If NoSwitch is FALSE or omitted, VDB automatically switches to straight-line depreciation when that is greater than declining-balance depreciation. If NoSwitch is TRUE, VDB never switches to straight-line depreciation.

6.12.51 XIRR

Summary: Compute the internal rate of return for a non-periodic series of cash flows.

Syntax: XIRR( NumberSequence Values ; DateSequence Dates [ ; Number Guess = 0.1 ] )

Constraints: The size of Values and Dates are equal. Values contains at least one positive and one negative cash flow.

Semantics: Compute the internal rate of return for a series of cash flows which is not necessarily periodic. The parameters are:

•Values: a series of cash flows. The first cash-flow amount is a negative number that represents the investment. The later cash flows are discounted based on the annual discount rate and the timing of the flow. The series of cash flow should contain at least one positive and one negative value.
•Dates: a series of dates that corresponds to values. The first date indicates the start of the cash flows. The range of Values and Dates shall be the same size.
•Guess: If provided, Guess is an estimate of the interest rate to start the iterative computation. If omitted, the value 0.1 (10%) is assumed. The result of XIRR is the rate at which the XNPV() function will return zero with the given cash flows. There is no closed form for XIRR. Implementations may return an approximate solution using an iterative method, in which case the Guess parameter may be used to initialize the iteration. If the implementation is unable to converge on a solution given a particular Guess, it may return an error.

6.12.52 XNPV

Summary: Compute the net present value of a series of cash flows.

Syntax: XNPV( Number Rate ; Reference|Array Values ; Reference|Array Dates )

Number of elements in Values equals number of elements in Dates.

All elements of Values are of type Number.

All elements of Dates are of type Number.

All elements of Dates ≥ Dates[1]

Semantics: Compute the net present value for a series of cash flows which is not necessarily periodic. The parameters are:

•Rate: discount rate. The value should be greater than -1.
•Values: a series of cash flows. The first cash-flow amount is a negative number that represents the investment. The later cash flows are discounted based on the annual discount rate and the timing of the flow. The series of cash flow should contain at least one positive and one negative value.
•Dates: a series of dates that corresponds to values. The first date indicates the start of the cash flows. If the dimensions of the Values and Dates arrays differ, evaluators shall match value and date pairs row-wise starting from top left.

With N being the number of elements in Values and Dates each, the formula is:

XNPV = \sum_{i = 1}^{N} \frac{{Values}_{i}}{{(1 + Rate)}^{\frac{{Dates}_{i} - {Dates}_{1}}{365}}}

6.12.53 YIELD

Summary: Calculate the yield of a bond.

Syntax: YIELD( DateParam Settlement ; DateParam Maturity ; Number Rate ; Number Price ; Number Redemption ; Number Frequency [ ; Basis B = 0 ] )

Constraints: Rate, Price, and Redemption should be greater than 0.

Summary: Calculate the yield of a discounted security per 100 currency units of face value.

•Settlement: the settlement/purchase date of the bond.
•Maturity: the maturity/expiry date of the bond.
•Rate: the interest rate of the bond.
•Price: the price of the bond per 100 currency units face value.
•Redemption: the redemption value of the bond per 100 currency units face value.
•Frequency: the number of interest payments per year. 1 = annual; 2 = semiannual; 4 = quarterly.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also PRICE 6.12.38, YIELDDISC 6.12.54, YIELDMAT 6.12.55

6.12.54 YIELDDISC

Syntax: YIELDDISC( DateParam Settlement ; DateParam Maturity ; Number Price ; Number Redemption [ ; Basis B = 0 ] )

Constraints: Price and Redemption should be greater than 0.

Summary: Calculate the yield of the security that pays interest on the maturity date.

•Settlement: the settlement/purchase date of the security.
•Maturity: the maturity/expiry date of the security.
•Price: the price of the security per 100 currency units face value.
•Redemption: the redemption value per 100 currency units face value.
•B: indicates the day-count convention to use in the calculation. 4.11.7

The return value is

YIELDDISC = \frac{\frac{Redemption}{Price} - 1}{YEARFRAC (Settlement; Maturity; Basis)}

See also PRICEDISC 6.12.39, YEARFRAC 6.10.24

6.12.55 YIELDMAT

Syntax: YIELDMAT( DateParam Settlement ; DateParam Maturity ; DateParam Issue ; Number Rate ; Number Price [ ; Basis B = 0 ] )

Constraints: Rate and Price should be greater than 0.

Information functions provide information about a data value, the spreadsheet, or underlying environment, including special functions for converting between data types.

•Settlement: the settlement/purchase date of the security.
•Maturity: the maturity/expiry date of the security.
•Issue: the issue date of the security.
•Rate: the interest rate of the security.
•Price: the price of the security per 100 currency units face value.
•B: indicates the day-count convention to use in the calculation. 4.11.7

See also PRICE 6.12.38, YIELD 6.12.53, YIELDDISC 6.12.54

6.13 Information Functions

6.13.1 General

6.13.2 AREAS

Summary: Returns the number of areas in a given list of references.

Syntax: AREAS( ReferenceList R )

Semantics: Returns the number of areas in the reference list R.

6.13.3 CELL

Summary: Returns information about position, formatting or contents in a reference.

Syntax: CELL( Text Info_Type [ ; Reference R ] )

Returns: Information about position, formatting properties or content

Summary: Returns the column number(s) of a reference.

•Info_Type: the text string which specifies the type of information. Please refer to Table 17 - CELL.
•R : if R is a reference to a cell, it is the cell whose information will be returned; if R is a reference to a range, the top-left cell in the range is the selected one; if R is omitted, the current cell is used.

Table 17 - CELL

Info_Type	Comment
COL	Returns the column number of the cell.
ROW	Returns the row number of the cell.
SHEET	Returns the sheet number of the cell.
ADDRESS	Returns the absolute address of the cell. The sheet name is included if given in the reference and does not reference the same sheet as the sheet the expression is evaluated upon. For an external reference a Source as specified in the syntax rules for References 5.8 is included.
FILENAME	Returns the file name of the file that contains the cell as an IRI. If the file is newly created and has not yet been saved, the file name is empty text “”.
CONTENTS	Returns the contents of the cell, without formatting properties.
COLOR	Returns 1 if color formatting is set for negative value in this cell; otherwise returns 0
FORMAT	Returns a text string which shows the number format of the cell. ,(comma) = number with thousands separator F = number without thousands separator C = currency format S = exponential representation P = percentage To indicate the number of decimal places after the decimal separator, a number is given right after the above characters. D1 = MMM-D-YY, MM-D-YY and similar formats D2 = DD-MM D3 = MM-YY D4 = DD-MM-YYYY HH:MM:SS D5 = MM-DD D6 = HH:MM:SS AM/PM D7 = HH:MM AM/PM D8 = HH:MM:SS D9 = HH:MM G = All other formats - (Minus) at the end = negative numbers in the cell have color setting () (brackets) at the end = this cell has the format settings with parentheses for positive or all values
TYPE	Returns the text value corresponding to the type of content in the cell: “b” : blank or empty cell content “l” : label or text cell content “v” : number value cell content
WIDTH	Returns the column width of the cell. The unit is the width of one zero (0) character in default font size.
PROTECT	Returns the protection status of the cell: 1 = cell is protected 0 = cell is unprotected
PARENTHESES	Returns 1 if the cell has the format settings with parentheses for positive or all values, otherwise returns 0
PREFIX	Returns single character text strings corresponding to the alignment of the cell. “'” (APOSTROPHE, U+0027) = left alignment '"' (QUOTATION MARK, U+0022) = right alignment “^” (CIRCUMFLEX ACCENT, U+005E) = centered alignment “\” (REVERSE SOLIDUS, U+005C) = filled alignment otherwise, returns empty string "".

6.13.4 COLUMN

Syntax: COLUMN( [ Reference R ] )

Constraints: AREAS(R) = 1

Semantics: Returns the column number of a reference, where “A” is 1, “B” is 2, and so on. If no parameter is given, the current cell is used. If a reference has multiple columns, an array of numbers is returned with all of the columns in the reference.

See also AREAS 6.13.2, ROW 6.13.29, SHEET 6.13.31

6.13.5 COLUMNS

Summary: Returns the number of columns in a given range.

Syntax: COLUMNS( Reference|Array R )

Semantics: Returns the number of columns in the range or array specified. The result is not dependent on the cell content in the range.

6.13.6 COUNT

Summary: Count the number of Numbers provided.

Syntax: COUNT( { NumberSequenceList N }+ )

Constraints: One or more parameters.

Semantics: Counts the numbers in the list N. Only numbers in references are counted; all other types are ignored. Errors are not propagated. It is implementation-defined what happens if 0 parameters are passed, but it should be an Error or 0.

6.13.7 COUNTA

Summary: Count the number of non-empty values.

Syntax: COUNTA( { Any AnyValue }+ )

Semantics: Counts the number of non-blank values. A value is non-blank if it contains any content of any type, including an Error. In a reference, every cell that is not empty is included in the count. An empty string value ("") is not considered blank. Errors contained in a range are considered a non-blank value for purposes of the count; errors do not propagate. Constant expressions or formulas are allowed; these are evaluated and if they produce an Error value the Error value is counted as one non-blank value (and not propagated as an Error). It is implementation-defined what happens if 0 parameters are passed, but it should be an Error or 0.

6.13.8 COUNTBLANK

Summary: Count the number of blank cells.

Syntax: COUNTBLANK( ReferenceList R )

Semantics: Counts the number of blank cells in R. A cell is blank if the cell is empty for purposes of COUNTBLANK. If ISBLANK(R) is TRUE, then it is blank. A cell with numeric value zero ('0') is not blank. It is implementation-defined whether or not a cell returning the empty string ("") is considered blank; because of this, there is a (potential) subtle difference between COUNTBLANK and ISBLANK.

Evaluators shall support one Reference as a parameter and may support a ReferenceList as a parameter.

See also COUNT 6.13.6, COUNTA 6.13.7, COUNTIF 6.13.9, ISBLANK 6.13.14

6.13.9 COUNTIF

Summary: Count the number of cells in a range that meet a criteria.

Syntax: COUNTIF( ReferenceList R ; Criterion C )

Constraints: Does not accept constant values as the reference parameter.

Semantics: Counts the number of cells in the reference range R that meet the Criterion C (4.11.8).

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also COUNT 6.13.6, COUNTA 6.13.7, COUNTBLANK 6.13.8, COUNTIFS 6.13.10, SUMIF 6.16.62, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.13.10 COUNTIFS

Summary: Count the number of cells that meet multiple criteria in multiple ranges.

Syntax: COUNTIFS( Reference R1 ; Criterion C1 [ ; Reference R2 ; Criterion C2 ]... )

Constraints: Does not accept constant values as the reference parameter.

Semantics: Counts the number of cells that meet the Criterion C1 in the reference range R1 and the Criterion C2 in the reference range R2, and so on (4.11.8). All reference ranges shall have the same dimension and size, else an Error is returned. A logical AND is applied between each array result of each selection; an entry is counted only if the same position in each array is the result of a Criterion match.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also AVERAGEIFS 6.18.6, COUNT 6.13.6, COUNTA 6.13.7, COUNTBLANK 6.13.8, COUNTIF 6.13.9, SUMIF 6.16.62, SUMIFS 6.16.63, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.13.11 ERROR.TYPE

Summary: Returns Number representing the specific Error type .

Syntax: ERROR.TYPE( Error E )

Semantics: Returns a number representing what kind of Error has occurred. Note that unlike most functions, this function does not propagate Error values. Receiving a non-Error value returns an Error. In particular, ERROR.TYPE(NA()) returns 7, and ERROR.TYPE applied to a non-Error returns an Error.

6.13.12 FORMULA

Summary: Returns formula at given reference as text.

Syntax: FORMULA( Reference X )

Returns: String

Constraints: Reference X shall contain a formula.

Semantics: Returns the formula in reference X as a string. The specific syntax of this returned string is implementation-defined. This function is intended to aid debugging by simplifying display of formulas in other cells. Error results of the referred formula cell are not propagated.

6.13.13 INFO

Summary: Returns information about the environment.

Syntax: INFO( Text Category )

Returns: Any (see below)

Constraints: Category shall be valid.

Semantics: Returns information about the environment in the given category.

Evaluators shall support at least the following categories:

Table 18 - INFO

Category	Meaning	Type
"directory"	Current directory. This shall be formatted so file names can be appended to the result (e.g., on POSIX and Windows systems it shall end with the separator “/” or “\” respectively).	Text
"memavail"	Amount of memory “available”, in bytes. On many modern (virtual memory) systems this value is not really available, but a system should return 0 if it is known that there is no more memory available, and greater than 0 otherwise	Number
"memused"	Amount of memory used, in bytes, by the data	Number
"numfile"	Number of active worksheets in files	Number
"osversion"	Operating system version	Text
"origin"	The top leftmost visible cell's absolute reference prefixed with “$A:”. In locales where cells are ordered right-to-left, the top rightmost visible cell is used instead.	Text
"recalc"	Current recalculation mode. If the locale is English, this is either "Automatic" or "Manual" (the exact text depends on the locale)	Text
"release"	The version of the implementation.	Text
"system"	The type of the operating system.	Text
"totmem"	Total memory available in bytes, including the memory already used.	Number

Evaluators may support other categories.

6.13.14 ISBLANK

Summary: Return TRUE if the referenced cell is blank, else return FALSE.

Syntax: ISBLANK( Scalar X )

Semantics: If X is of type Number, Text, or Logical, return FALSE. If X is a reference to a cell, examine the cell; if it is blank (has no value), return TRUE, but if it has a value, return FALSE. A cell with the empty string is not considered blank. This function does not propagate Error values.

6.13.15 ISERR

Summary: Return TRUE if the parameter has type Error and is not #N/A, else return FALSE.

Syntax: ISERR( Scalar X )

Summary: Return TRUE if the parameter has type Error, else return FALSE.

Semantics: If X is of type Error, and ISNA(X) is not true, returns TRUE. Otherwise it returns FALSE. Note that this function returns FALSE if given #N/A; if this is not desired, use ISERROR 6.13.16. Note that this function does not propagate Error values.

ISERR(X) is the same as:

IF(ISNA(X),FALSE(),ISERROR(X))

See also ERROR.TYPE 6.13.11, ISERROR 6.13.16, ISNA 6.13.20, ISNUMBER 6.13.22, ISTEXT 6.13.25, NA 6.13.27

6.13.16 ISERROR

Syntax: ISERROR( Scalar X )

Summary: Return TRUE if the value is even, else return FALSE.

Semantics: If X is of type Error, returns TRUE, else returns FALSE. Note that this function returns TRUE if given #N/A; if this is not desired, use ISERR 6.13.15. Note that this function does not propagate Error values.

See also ERROR.TYPE 6.13.11, ISERR 6.13.15, ISNA 6.13.20, ISNUMBER 6.13.22, ISTEXT 6.13.25, NA 6.13.27

6.13.17 ISEVEN

Syntax: ISEVEN( Number X )

Semantics: First, compute X1 = TRUNC(X). Then, if X1 is even (a division by 2 has a remainder of 0), return TRUE, else return FALSE. The result is implementation-defined if given a Logical value; an evaluator may return either an Error or the result of converting the Logical value to a Number (per Conversion to Number 6.3.5 ).

6.13.18 ISFORMULA

Summary: Return TRUE if the reference refers to a formula, else return FALSE.

Syntax: ISFORMULA( Reference X )

Semantics: If X refers to a cell whose value is computed by a formula, return TRUE, else return FALSE. A formula itself may compute a constant; in that case it will still return TRUE since it is still a formula. Passing a non-reference, or a reference to more than one cell, is implementation-defined. This function does not propagate Error values.

6.13.19 ISLOGICAL

Summary: Return TRUE if the parameter has type Logical, else return FALSE.

Syntax: ISLOGICAL( Scalar X )

Semantics: If X is of type Logical, returns TRUE, else FALSE. Evaluators that do not have a distinct Logical type will return the same value ISNUMBER(X) would return. This function does not propagate Error values.

6.13.20 ISNA

Summary: Return TRUE if the parameter has type Error and is #N/A, else return FALSE.

Syntax: ISERR( Scalar X )

Semantics: If X is #N/A, return TRUE, else return FALSE. Note that if X is a reference, the value being referenced is considered. This function does not propagate Error values.

See also ERROR.TYPE 6.13.11, ISERROR 6.13.16, ISERR 6.13.15, ISNUMBER 6.13.22, ISTEXT 6.13.25, NA 6.13.27

6.13.21 ISNONTEXT

Summary: Return TRUE if the parameter does not have type Text, else return FALSE.

Syntax: ISNONTEXT( Scalar X )

Summary: Return TRUE if the parameter has type Number, else return FALSE.

Semantics: If X is of type Text, ISNONTEXT returns FALSE, else TRUE. If X is a reference, it examines what X references. References to empty cells are not considered text, so for reference to an empty cell ISNONTEXT will return TRUE. Empty Cell 4.7 This function does not propagate Error values.

ISNONTEXT(X) is equivalent to NOT(ISTEXT(X))

See also ISNUMBER 6.13.22, ISLOGICAL 6.13.19, ISTEXT 6.13.25, NOT 6.15.7

6.13.22 ISNUMBER

Syntax: ISNUMBER( Scalar X )

Semantics: If X is of type Number, returns TRUE, else FALSE. Evaluators need not have a distinguished Logical type; in such evaluators, ISNUMBER(TRUE()) is TRUE. This function does not propagate Error values.

See also ISTEXT 6.13.25, ISLOGICAL 6.13.19

6.13.23 ISODD

Summary: Return TRUE if the value is even, else return FALSE.

Syntax: ISODD( Number X )

Semantics: First, compute X1 = TRUNC(X). Then, if X1 is odd (a division by 2 has a remainder of 1), return TRUE, else return FALSE. The result is implementation-defined if given a Logical value; an evaluator may return either an Error or the result of converting the Logical value to a Number (per Conversion to Number 6.3.5 ).

6.13.24 ISREF

Summary: Return TRUE if the parameter is of type reference, else return FALSE.

Syntax: ISREF( Any X )

Semantics: If X is of type Reference or ReferenceList, return TRUE, else return FALSE. Note that unlike nearly all other functions, when given a reference this function does not then examine the value being referenced. Some functions and operators return references, and thus ISREF will return TRUE when given their results. X may be a ReferenceList, in which case ISREF returns TRUE. This function does not propagate Error values.

6.13.25 ISTEXT

Summary: Return TRUE if the parameter has type Text, else return FALSE.

ISTEXT(X) is equivalent to NOT(ISNONTEXT(X)).

Syntax: ISTEXT( Scalar X )

Summary: Return the number of a value.

Semantics: If X is of type Text, returns TRUE, else FALSE. References to empty cells are NOT considered Text. If X is a reference, examines what X references. References to empty cells are NOT considered Text, so a reference to an empty cell will return FALSE. Empty Cell 4.7 This function does not propagate Error values.

See also ISNONTEXT 6.13.21, ISNUMBER 6.13.22, ISLOGICAL 6.13.19

6.13.26 N

Syntax: N( Any X )

Semantics: If X is a Reference, it is first dereferenced to a scalar. Then its type is examined. If it is of type Number, it is returned. If it is of type Logical, 1 is returned if TRUE else 0 is returned. It is implementation-defined what happens if it is provided a Text value.

6.13.27 NA

Summary: Return the constant Error value #N/A.

Syntax: NA()

Returns: Error

Constraints: Shall have 0 parameters

Semantics: This function takes no arguments and returns the Error #N/A.

6.13.28 NUMBERVALUE

Summary: Convert text to number, in a locale-independent way.

Syntax: NUMBERVALUE( Text X [ ; Text DecimalSeparator [ ; Text GroupSeparator ] ] )

Constraints: LEN(DecimalSeparator) = 1, DecimalSeparator shall not appear in GroupSeparator

Semantics: Converts given Text value X into Number. If X is a Reference, it is first dereferenced.

X is transformed according to the following rules:

1.Starting from the beginning, remove all occurrences of the group separator before any decimal separator
2.Starting from the beginning, replace the first occurrence in the text of the decimal separator character with the FULL STOP (U+002E) character
3.Remove all whitespace characters (5.14).
4.If the first character of the resulting string is a period FULL STOP (U+002E) then prepend a zero
5.If the string ends in one or more instances of PERCENT SIGN (U+0025) , remove the percent sign(s)

If percent signs were removed in step 5, divide the value of the returned number by 100 for each percent sign removed. If the resulting string is a valid xsd:float, then return the number corresponding to that string, according to the definition provided in XML Schema, Part 2, Section 3.2.4.

If the string is not a valid xsd:float then return an error.

See also N 6.13.26, T 6.20.22, DATEVALUE 6.10.4, TIMEVALUE 6.10.18, VALUE 6.13.34

6.13.29 ROW

Summary: Returns the row number(s) of a reference.

Syntax: ROW( [ Reference R ] )

Constraints: AREAS(R) = 1

Semantics: Returns the row number of a reference. If no parameter is given, the current cell is used. If a reference has multiple rows, an array of numbers is returned with all of the rows in the reference.

See also AREAS 6.13.2, COLUMN 6.13.4, SHEET 6.13.31

6.13.30 ROWS

Summary: Returns the number of rows in a given range.

Syntax: ROWS( Reference|Array R )

Semantics: The result is not dependent on the cell content in the range.

6.13.31 SHEET

Summary: Returns the sheet number of the reference or the string representing a sheet name.

Syntax: SHEET( [ Text|Reference R ] )

Returns: Number ≥ 1

Constraints: R shall not contain a Source Location (5.8 References)

Semantics: Returns the 1-based sheet number of the given reference or sheet name.

Hidden sheets are not excluded from the sheet count.

If no parameter is given, the result is the sheet number of the sheet containing the formula.

If a Reference is given it is not dereferenced.

If the reference encompasses more than one sheet, the result is the number of the first sheet in the range.

If a reference does not contain a sheet reference, the result is the sheet number of the sheet containing the formula.

If the function is not evaluated within a table cell, an error is returned.

See also COLUMN 6.13.4, ROW 6.13.29, SHEETS 6.13.32

6.13.32 SHEETS

Summary: Returns the number of sheets in a reference or current document.

Syntax: SHEETS( [ Reference R ] )

Returns: Number ≥ 1

Constraints: R shall not contain a Source Location (5.8 References)

Semantics: Returns the number of sheets in the given reference.

If no parameter is given, the number of sheets in the document is returned.

Hidden sheets are not excluded from the sheet count.

See also COLUMNS 6.13.5, ROWS 6.13.30, SHEET 6.13.31

6.13.33 TYPE

Summary: Returns a number indicating the type of the provided value.

Syntax: TYPE( Any Value )

Semantics: Returns a number indicating the type of the value given:

Table 19 - TYPE

Value's Type	TYPE Return
Number	1
Text	2
Logical	4
Error	16
Array	64

If a Reference is provided, the reference is first dereferenced, and any formulas are evaluated. This function does not propagate Error values.

Note: Reliance on the return of 4 for TYPE will impair the interoperability of a document containing an expression that relies on that value.

6.13.34 VALUE

Summary: Convert text to number.

Syntax: VALUE( Text X )

Semantics: Converts given text value X into Number. If X is a Reference, it is first dereferenced. It is implementation-defined what happens if VALUE is given neither a Text value nor a Reference to a Text value. If the Text has a date, time, or datetime format, it is converted into a serial Number. In many cases the conversion of a date or datetime format is locale-dependent.

If the supplied text X cannot be converted into a Number, an Error is returned.

Regardless of the current locale, an evaluator shall accept numbers matching this regular expression (which does not include a decimal point character) and convert it into a Number. If the value ends in %, it shall divide the number by 100:

[+-]? [0-9]+([eE][+-]?[0-9]+)?)%?

VALUE shall accept text representations of numbers in the current locale. In the en_US locale, an evaluator shall accept decimal numbers matching this regular expression and convert it into a Number (the leading “$” is ignored; commas are ignored if they match the rule of a thousands separator; if the value ends in %, it shall divide the number by 100):

[+-]?\$?([0-9]+(,[0-9]{3})*)?(\.[0-9]+)?(([eE][+-]?[0-9]+)|%)?

Evaluators shall accept fractional values matching the regular expression:

[+-]? [0-9]+ \ [0-9]+/[1-9][0-9]?

A leading minus sign shall be interpreted as identifying a negative number for the entire value. There is a space between the integer and the fractional portion; values between 0 and 1 can be represented by using 0 for the integer part.

Evaluators shall support time values in at least the HH:MM and HH:MM:SS formats, where HH is a 1-2 digit value from 0 to 23, MM is a one- or two-digit value from 0 to 59, and SS is a one- or two-digit value from 0 to 59. The hour may be one or two digits when it is less than 10. VALUE converts time values into Numbers ranging from 0 to 1, which is percentage of day that has elapsed by that time. Thus, VALUE("2:00") is the same as 2/24. Evaluators should accept times with fractional seconds as well when expressed in the form HH:MM:SS.ssss...

Evaluators shall accept textual dates in [ISO8601] format (YYYY-MM-DD), converting them into serial numbers based on the current epoch. Evaluators shall, when running in the en_US locale, accept the format MM/DD/YYYY .

In addition, in locale en_US, evaluators shall support the following formats (where YYYY is a 4-digit year, YY a 2-digit year, MM a numerical month, DD a numerical day, mmm a 3-character abbreviated alphabetical name, and mmmmm a full name):

Table 20 - VALUE

Format	Example	Comment
MM/DD/YYYY	5/21/2006	LOCALE-DEPENDENT; Long year format with slashes.
MM/DD/YY	5/21/06	LOCALE-DEPENDENT; Short year format with slashes
MM-DD-YYYY	5-21-2006	LOCALE-DEPENDENT; Long year format with dashes (short year may be supported, but it may also be used for years less than 100 .
mmm DD, YYYY	Oct 29, 2006	LOCALE-DEPENDENT; Short alphabetic month day, year. Note: mmm depends on the locale's language.
DD mmm YYYY	29 Oct 2006	LOCALE-DEPENDENT; Short alphabetic day month year
mmmmm DD, YYYY	October 29, 2006	LOCALE-DEPENDENT; Long alphabetic month day, year
DD mmmmm YYYY	29 October 2006	LOCALE-DEPENDENT; Long alphabetic day month year

Evaluators should support other locales. Many conversions will vary by locale, including the decimal point (comma or period), names of months, date formats (MM/DD vs. DD/MM), and so on. Dates in particular vary by locale.

Evaluators shall support the datetime format, which is a date followed by a time, using either the space character or the literal “T” character as the separator (the “T” is from ISO 8601). Evaluators shall support at least the ISO date format in a datetime format; they may support other date formats in a datetime format as well. Formats such as “YYYY-MM-DD HH:MM” and “YYYY-MM-DDTHH:MM:SS” (where “T” is the literal character T) shall be accepted. The result of accepting a datetime format shall be a representation of that specific time (without removing either the date or the time of day, unlike DATEVALUE or TIMEVALUE).

Evaluators may accept other formats that will convert to numbers, and those conversions may be locale-dependent, as long as they do not conflict with the above. Where no conversion is determined, an Error is returned.

See also N 6.13.26, T 6.20.22, DATEVALUE 6.10.4, TIMEVALUE 6.10.18, NUMBERVALUE 6.13.28

6.14 Lookup Functions

6.14.1 General

These functions look up information. Note that IF() 6.15.4 can be considered a trivial lookup function, but it is listed as a logical function instead.

6.14.2 ADDRESS

Summary: Returns a cell address (reference) as text.

Syntax: ADDRESS( Integer Row ; Integer Column [ ; Integer Abs = 1 [ ; Logical A1Style = TRUE [ ; Text Sheet ] ] ] )

Constraints: Row ≥ 1, Column ≥ 1, 1 ≤ Abs ≤ 4; A1Style = TRUE. Evaluators may evaluate expressions that do not meet the constraint A1Style = TRUE.

Semantics: Returns a cell address (reference) as text. The text does not include the surrounding [...] of a reference. If a Sheet name is given, the sheet name in the text returned is followed by a “.” and the column/row reference if A1Style is TRUE, or a “!” and the column/row reference if A1Style is FALSE; otherwise no “.” respectively “!” is included. Columns are identified using uppercase letters. The value of Abs determines if the column and/or row is absolute or relative. The value of A1Style determines if A1 reference style or R1C1 reference style is used.

Table 21 - ADDRESS

Abs	Meaning	A1Style = TRUE	A1Style = FALSE
1	fully absolute	$A$1	R1C1
2	row absolute, column relative	A$1	R1C[1]
3	row relative, column absolute	$A1	R[1]C1
4	fully relative	A1	R[1]C[1]

Note that the INDIRECT function accepts this format.

6.14.3 CHOOSE

Summary: Uses an index to return a value from a list of values.

Syntax: CHOOSE( Integer Index ; { Any Value }+ )

Constraints: Returns an Error if Index < 1 or if there is no corresponding value in the list of Values.

Semantics: Uses Index to determine which value, from a list of values, to return. If Index is 1, CHOOSE returns the first Value; if Index is 2, CHOOSE returns the second value, and so on. Note that the Values may be formula expressions. Expression paths of parameters other than the one chosen are not calculated or evaluated for side effects.

6.14.4 GETPIVOTDATA

6.14.4.1 General

Summary: Return a value from a data pilot table.

Note: This function has two syntaxes. The first of these is the preferred syntax, while the second, alternative syntax is provided for compatibility reasons.

6.14.4.2 Preferred Syntax

Syntax: GETPIVOTDATA( Text DataField ; Reference Table { ; Text FieldName ; Scalar Member }* )

Note: This version of the syntax is distinguished by the parameter Table being the second parameter.

Returns: Any

Semantics: Returns a single result from the calculation of a data pilot table.

The data pilot table is selected by Table, which is a reference to a cell or cell range that's within a data pilot table or contains a data pilot table. If the cell range contains several data pilot tables, the last one in the order of <table:data-pilot-table> elements (OpenDocument, Part 3, 9.6.3) in the file is used.

DataField selects one of the data pilot table's data fields. It can be the name of the source column, or the given name of the data field (such as “Sum of Sales”).

If no FieldName/Member pairs are given, the grand total is returned. Otherwise, each pair adds a constraint that the result shall satisfy. FieldName is the name of a field from the data pilot table. Member is the name of a member (item) from that field. If a member is a number, Member can alternatively be its numerical value.

If the data pilot table contains only a single result value that fulfills all of the constraints, or a subtotal result that summarizes all matching values, that result is returned. If there is no matching result, or several ones without a subtotal for them, an Error is returned. These conditions apply to results that are included in the data pilot table. If the source data contains entries that are hidden by settings of the data pilot table, they are ignored. The order of the FieldName/Member pairs is not significant. Field and member names are case-insensitive.

If no constraint for a page field is given, the field's selected value is implicitly used. If a constraint for a page field is given, it shall match the field's selected value, or an Error is returned.

Subtotal values from the data pilot table are only used if they use the function “auto” (except when specified in the constraint, see below).

6.14.4.3 Alternative Syntax

Syntax: GETPIVOTDATA( Reference Table ; Text Constraints )

Note: This version of the syntax is distinguished by the parameter Table being the first parameter.

Returns: Any

Semantics: Returns a single result from the calculation of a data pilot table.

Table has the same meaning as in the preferred syntax.

Constraints is a space-separated list. Entries can be quoted (single quotes). One of the entries can be the data field name. The data field name can be left out if the data pilot table contains only one data field, otherwise it shall be present. Each of the other entries specifies a constraint in the form FieldName[Member] (with literal characters “[“ (LEFT SQUARE BRACKET, U+005B) and “]” (RIGHT SQUARE BRACKET U+005D)), or only Member if the member name is unique within all fields that are used in the data pilot table. A function name can be added in the form FieldName[Member;Function], which will cause the constraint to match only subtotal values which use that function. The possible function names are the same as in the table:function attribute of the <table:data-pilot-subtotal> element (OpenDocument, Part 3, 19.647.4), case-insensitive.

6.14.5 HLOOKUP

Summary: Look for a matching value in the first row of the given table, and return the value of the indicated row.

Syntax: HLOOKUP( Any Lookup ; ForceArray Reference|Array DataSource ; Integer Row [ ; Logical RangeLookup = TRUE ] )

Constraints: Row ≥ 1; Searched portion of DataSource shall not include Logical values. Evaluators may evaluate expressions that do not meet the constraint that the searched portion of a DataSource not include Logical values.

If RangeLookup is omitted or TRUE or not 0, the first row of DataSource is assumed to be sorted in ascending order, with smaller numbers before larger ones, smaller text values before larger ones (e.g., "A" before "B", and "B" before "BA"), and FALSE before TRUE. If the types are mixed, Numbers are sorted before Text, and Text before Logicals; evaluators without a separate Logical type may include a Logical as a Number. The lookup will try to match an entry of value Lookup. If none is found the largest entry less than Lookup is taken as a match. From a sequence of identical values ≤ Lookup the last entry is taken. If there is no data less than or equal to Lookup, the #N/A Error is returned. If Lookup is of type Text and the value found is of type Number, the #N/A Error is returned. If DataSource is not sorted, the result is undetermined and implementation-dependent. In most cases it will be arbitrary and just plain wrong due to binary search algorithms.

If RangeLookup is FALSE or 0, DataSource does not need to be sorted and an exact match is searched. Each value in the first row of DataSource is examined in order (starting at the left) until its value matches Lookup.

Both methods, if there is a match, return the corresponding value in row Row, relative to the DataSource, where the topmost row in DataSource is 1.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also INDEX 6.14.6, MATCH 6.14.9, OFFSET 6.14.11, VLOOKUP 6.14.12

6.14.6 INDEX

Summary: Returns a value using a row and column index value (and optionally an area index).

Syntax: INDEX( ReferenceList|Array DataSource ; [ Integer Row ] [ ; [ Integer Column ] ] [ ; Integer AreaNumber = 1 ] )

Constraints: Row ≥ 0, Column ≥ 0,
1 ≤ AreaNumber ≤ number of references in DataSource if that is a ReferenceList, else AreaNumber = 1

Given a DataSource, returns the value at the given Row and Column (starting numbering at 1, relative to the top-left of the DataSource) of the given area AreaNumber. If AreaNumber is not given, it defaults to 1 (the first and possibly only area). This function is essentially a two-dimensional version of CHOOSE, which does not accept range parameters.

If Row is omitted or an empty parameter (two consecutive ;; semicolons) or 0, an entire column of the given area AreaNumber in DataSource is returned. If Column is omitted or an empty parameter (two consecutive ;; semicolons) or 0, an entire row of the given area AreaNumber in DataSource is returned. If both, Row and Column, are omitted or empty or 0, the entire given area AreaNumber is returned.

If DataSource is a one-dimensional column vector, Column is optional or can be omitted as an empty parameter (two consecutive ;; semicolons). If DataSource is a one-dimensional row vector, Row is optional, which effectively makes Row act as the column offset into the vector, or can be omitted as an empty parameter (two consecutive ;; semicolons).

If Row or Column have a value greater than the dimension of the corresponding given area AreaNumber, an Error is returned.

6.14.7 INDIRECT

Summary: Return a reference given a string representation of a reference.

Syntax: INDIRECT( Text Ref [ ; Logical A1 = TRUE ] )

Constraints: Ref is valid reference

Semantics: Given text for a reference (such as “A3”), returns a reference. If A1 is False, it is interpreted as an R1C1 reference style. For interoperability, if the Ref text includes a sheet name, evaluators should be able to parse both, the “.” dot and the “!” exclamation mark, as the sheet name separator. If evaluators support the A1 = FALSE case of the ADDRESS 6.14.2 function and include the “!” exclamation mark as the sheet name separator, evaluators shall correctly parse that in the A1 = FALSE case of this INDIRECT function. Evaluators shall correctly parse the “.” dot as the sheet name separator in the A1 = TRUE case.

6.14.8 LOOKUP

Summary: Look for criterion in an already-sorted array, and return a corresponding result.

Syntax: LOOKUP( Any Find ; ForceArray Reference|Array Searched [ ; ForceArray Reference|Array Results ] )

Constraints: The searched portion of Searched shall be sorted in ascending order; if provided, Results shall have the same length as Searched. The searched portion of Searched shall not include Logical values. Evaluators may evaluate expressions that do not meet the constraint that the searched portion of a Searched not include Logical values.

Semantics: This function searches for Find in a row or column of the previously-sorted array Searched and returns a corresponding value. The match is the largest value in the row/column of Searched that is less than or equal to Find (so an exact match is always preferred over inexact ones). From a sequence of identical values ≤ Find the last entry is taken. If Find is smaller than the smallest value in the first row or column (depending on the array dimensions), LOOKUP returns the #N/A Error. If Find is of type Text and the value found is of type Number, the #N/A Error is returned.

The searched portion of Searched shall be sorted in ascending order, and so that values of type Number precede values of type Text if both types are included (e.g., -2, 0, 2, “A”, “B”).

There are two major uses for this function; the 3-parameter version (vector) and the 2-parameter version (non-vector array).

Note: Interoperability is improved by use of HLOOKUP or VLOOKUP in expressions over LOOKUP.

When given two parameters, Searched is first examined:

•If Searched is square or is taller than it is wide (more rows than columns), LOOKUP searches in the first column (similar to VLOOKUP), and returns the corresponding value in the last column.
•If Searched covers an area that is wider than it is tall (more columns than rows), LOOKUP searches in the first row (similar to HLOOKUP), and returns the corresponding value in the last row.

When given 3 parameters, Results shall be a vector (either a row or a column) or an Error is raised. The function determines the index of the match in the first column respectively row of Searched, and returns the value in Results with the same index.

Searched is first examined:

•If Searched is square or is taller than it is wide (more rows than columns), LOOKUP searches in the first column (similar to VLOOKUP).
•If Searched covers an area that is wider than it is tall (more columns than rows), LOOKUP searches in the first row (similar to HLOOKUP).

The lengths of the search vector and the result vector do not need to be identical. When the match position falls outside the length of the result vector, an Error is returned if the result vector is given as an array object. If it is a cell range, it gets automatically extended to the length of the searched vector, but in the direction of the result vector. If just a single cell reference was passed, a column vector is generated. If the cell range cannot be extended due to the sheet's size limit, then the #N/A Error is returned.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also HLOOKUP 6.14.5, INDEX 6.14.6, MATCH 6.14.9, OFFSET 6.14.11, VLOOKUP 6.14.12

6.14.9 MATCH

Summary: Finds a Search item in a sequence, and returns its position (starting from 1).

Syntax: MATCH( Scalar Search ; ForceArray Reference|Array SearchRegion [ ; Integer MatchType = 1 ] )

Constraints: -1 ≤ MatchType ≤ 1; The searched portion of SearchRegion shall not include Logical values. Evaluators may evaluate expressions that do not meet the constraint that the searched portion of a SearchRegion not include Logical values.

SearchRegion shall be a vector (a single row or column)

If a match is found, MATCH returns the relative position (starting from 1). For Text the comparison is case-insensitive. MatchType determines the type of search; if MatchType is 0, the SearchRegion shall be considered unsorted, and the first match is returned. If MatchType is 1, the SearchRegion may be assumed to be sorted in ascending order, with smaller Numbers before larger ones, smaller Text values before larger ones (e.g., "A" before "B", and "B" before "BA"), and FALSE before TRUE. If the types are mixed, Numbers are sorted before Text, and Text before Logicals; evaluators without a separate Logical type may include a Logical as a Number. If MatchType is -1, then SearchRegion may be assumed to be sorted in descending order (the opposite of the above). If MatchType is 1 or -1, evaluators may use binary search or other techniques so that they do not need to examine every value in linear order. MatchType defaults to 1.

•MatchType = -1 finds the smallest value that is greater than or equal to Search in a SearchRegion where values are sorted in descending order. From a sequence of identical values ≥ Search the last value is taken. If no value ≥ Search exists, the #N/A Error is returned. If Search is of type Number and the value found is of type Text, the #N/A Error is returned.
•MatchType = 0 finds the first value that is equal to Search. Values in SearchRegion do not need to be sorted. If no value equal to Search exists, the #N/A Error is returned.
•MatchType = 1 or omitted finds the largest value that is less than or equal to Search in a SearchRegion where values are sorted in ascending order. From a sequence of identical values ≤ Search the last value is taken. If no value ≤ Search exists, the #N/A Error is returned. If Search is of type Text and the value found is of type Number, the #N/A Error is returned.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also HLOOKUP 6.14.5, OFFSET 6.14.11, VLOOKUP 6.14.12

6.14.10 MULTIPLE.OPERATIONS

Summary: Executes a formula expression while substituting a row reference and a column reference.

Syntax: MULTIPLE.OPERATIONS( Reference FormulaCell ; Reference RowCell ; Reference RowReplacement [ ; Reference ColumnCell ; Reference ColumnReplacement ] )

MULTIPLE.OPERATIONS executes the formula expression pointed to by FormulaCell and all formula expressions it depends on while replacing all references to RowCell with references to RowReplacement respectively all references to ColumnCell with references to ColumnReplacement.

•FormulaCell: reference to the cell that contains the formula expression to calculate.
•RowCell: reference that is to be replaced by RowReplacement.
•RowReplacement: reference that replaces RowCell.
•ColumnCell: reference that is to be replaced by ColumnReplacement.
•ColumnReplacement: reference that replaces ColumnCell.

If calls to MULTIPLE.OPERATIONS are encountered in dependencies, replacements of target cells shall occur in queued order, with each replacement using the result of the previous replacement.

Note: The function may be used to create tables of expressions that depend on two input parameters.

Example: FormulaCell is B5, RowCell is B3, ColumnCell is B2

Table 22 - MULTIPLE.OPERATIONS

	col_B	col_C	col_D	col_E	col_F
row_2	1		1	2	3
row_3	1	1	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C3;$B$2;F$2)
row_4	=B2+B3	2	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C4;$B$2;F$2)
row_5	=B2*B3+B4	3	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C5;$B$2;F$2)
		4	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;D$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;E$2)	=MULTIPLE.OPERATIONS($B$5;$B$3;$C6;$B$2;F$2)

Result:

Table 23 - MULTIPLE.OPERATIONS

	col_B	col_C	col_D	col_E	col_F
row_2	1		1	2	3
row_3	1	1	3	5	7
row_4	2	2	5	8	11
row_5	3	3	7	11	15
		4	9	14	19

Note that although only cell B5 is passed as the FormulaCell parameter, also the references to B2 and B3 of the formula in cell B4 are replaced, because B5 depends on B4.

6.14.11 OFFSET

Summary: Modifies a reference's position and dimension.

Syntax: OFFSET( Reference R ; Integer RowOffset ; Integer ColumnOffset [ ; [ Integer NewHeight ] [ ; [ Integer NewWidth ] ] ] )

Constraints: NewWidth > 0; NewHeight > 0
The modified reference shall be in a valid range, starting from column/row one to the maximum column/row.

Semantics: Shifts reference by RowOffset rows and by ColumnOffset columns. Optionally, the dimension can be set to NewWidth and/or NewHeight, if omitted the width/height of the original reference is taken. Note that NewHeight may be empty (two consecutive semicolons ;;) followed by a given NewWidth argument. Returns the modified reference.

See also COLUMN 6.13.4, COLUMNS 6.13.5, ROW 6.13.29, ROWS 6.13.30

6.14.12 VLOOKUP

Summary: Look for a matching value in the first column of the given table, and return the value of the indicated column.

Syntax: VLOOKUP( Any Lookup ; ForceArray Reference|Array DataSource ; Integer Column [ ; Logical RangeLookup = TRUE() ] )

Constraints: Column ≥ 1; The searched portion of DataSource shall not include Logical values. Evaluators may evaluate expressions that do not meet the constraint that the searched portion of a DataSource not include Logical values.

If RangeLookup is omitted or TRUE or not 0, the first column of DataSource is assumed to be sorted in ascending order, with smaller Numbers before larger ones, smaller Text values before larger ones (e.g., "A" before "B", and "B" before "BA"), and FALSE before TRUE. If the types are mixed, Numbers are sorted before Text, and Text before Logicals; evaluators without a separate Logical type may include a Logical as a Number. The lookup will try to match an entry of value Lookup. From a sequence of identical values ≤ Lookup the last entry is taken. If none is found the largest entry less than Lookup is taken as a match. If there is no data less than or equal to Lookup, the #N/A Error is returned. If Lookup is of type Text and the value found is of type Number, the #N/A Error is returned. If DataSource is not sorted, the result is undetermined and implementation-dependent. In most cases it will be arbitrary and just plain wrong due to binary search algorithms.

If RangeLookup is FALSE or 0, DataSource does not need to be sorted and an exact match is searched. Each value in the first column of DataSource is examined in order (starting at the top) until its value matches Lookup. If no value matches, the #N/A Error is returned.

Both methods, if there is a match, return the corresponding value in column Column, relative to the DataSource, where the leftmost column in DataSource is 1.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also HLOOKUP 6.14.5, INDEX 6.14.6, MATCH 6.14.9, OFFSET 6.14.11

6.15 Logical Functions

6.15.1 General

The logical functions are: TRUE() and FALSE(); the functions that compute Logical values NOT(), AND(), and OR(); and the conditional function IF(). The OpenDocument specification mentions "logical operators"; these are another name for the logical functions.

Note that because of Error values, any logical function that accepts parameters can actually produce TRUE, FALSE, or an Error value instead of TRUE or FALSE.

These are not bitwise operations, e.g., AND(12;10) produces TRUE, not 8. See the bit operation functions for bitwise operations.

6.15.2 AND

Summary: Compute logical AND of all parameters.

Syntax: AND( { Logical|NumberSequenceList L }+ )

Constraints: Shall have 1 or more parameters

Semantics: Computes the logical AND of the parameters. If all parameters are TRUE, returns TRUE; if any are FALSE, returns FALSE. When given one parameter, this has the effect of converting that one parameter into a Logical value. When given zero parameters, evaluators may return a Logical value or an Error.

Also in array context a logical AND of all arguments is computed, range or array parameters are not evaluated as a matrix and no array is returned. This behavior is consistent with functions like SUM. To compute a logical AND of arrays per element use the * operator in array context.

6.15.3 FALSE

Summary: Returns constant FALSE.

Syntax: FALSE()

Constraints: Shall have 0 parameters

Semantics: Returns logical constant FALSE. This may be a Number or a distinct type.

6.15.4 IF

Summary: Return one of two values, depending on a condition.

Syntax: IF( Logical Condition [ ; [ Any IfTrue ] [ ; [ Any IfFalse ] ] ] )

Semantics: Computes Condition. If it is TRUE, it returns IfTrue, else it returns IfFalse. This function only evaluates IfTrue, or IfFalse, and never both; that is to say, it short-circuits.

Seven versions are possible:

One parameter:

a)IF(Condition)

Two parameters:

b)IF(Condition;)
c)IF(Condition;IfTrue)

Three parameters:

d)IF(Condition;;)
e)IF(Condition;;IfFalse)
f)IF(Condition;IfTrue;)
g)IF(Condition;IfTrue;IfFalse)

If there is only 1 parameter (case a), IfTrue is considered to be TRUE and IfFalse is considered to be FALSE. Thus the 1 parameter version converts Condition into a Logical value.

If there are 2 parameters (cases b and c), IfFalse is considered to be FALSE. If there are 2 parameters and the second parameter is null (semicolon but no IfTrue, case b), IfTrue is considered to be 0.

If there are 3 parameters but the second parameter is null (two consecutive ;; semicolons, cases d and e), IfTrue is considered to be 0.

If there are 3 parameters but the third parameter is null (semicolon but no IfFalse, cases d and f), IfFalse is considered to be 0.

6.15.5 IFERROR

Summary: Return X unless it is an Error, in which case return an alternative value.

Syntax: IFERROR( Any X ; Any Alternative )

Semantics: Computes X. If ISERROR(X) is TRUE, return Alternative, else return X.

Note: This is semantically equivalent to IF(ISERROR(X); Alternative; X), except that X is only computed once.

6.15.6 IFNA

Summary: Return X unless it is #N/A, in which case return an alternative value.

Syntax: IFNA( Any X ; Any Alternative )

Semantics: Computes X. If ISNA(X) is TRUE, return Alternative, else return X.

Note: This is semantically equivalent to IF(ISNA(X); Alternative; X), except that X is only computed once.

6.15.7 NOT

Summary: Compute logical NOT.

Syntax: NOT( Logical L )

Constraints: Shall have 1 parameter.

Semantics: Computes the logical NOT. If given TRUE, returns FALSE; if given FALSE, returns TRUE.

6.15.8 OR

Summary: Compute logical OR of all parameters.

Syntax: OR( { Logical|NumberSequenceList L }+ )

Constraints: Shall have 1 or more parameters

Semantics: Computes the logical OR of the parameters. If all parameters are FALSE, it shall return FALSE; if any are TRUE, it shall returns TRUE. When given one parameter, this has the effect of converting that one parameter into a Logical value. When given zero parameters, evaluators may return a Logical value or an Error.

Also in array context a logical OR of all arguments is computed, range or array parameters are not evaluated as a matrix and no array is returned. This behavior is consistent with functions like SUM. To compute a logical OR of arrays per element use the + operator in array context.

6.15.9 TRUE

Summary: Returns constant TRUE

Syntax: TRUE()

Constraints: Shall have 0 parameters

Semantics: Returns logical constant TRUE. The result of this function may but need not be equal to 1 when compared using “=”. It always has the value of 1 if used in a context requiring Number (because of the automatic conversions), so if ISNUMBER(TRUE()) is TRUE, then it shall have the value 1.

See also FALSE 6.15.3, IF 6.15.4, ISNUMBER 6.13.22

6.15.10 XOR

Summary: Compute a logical XOR of all parameters.

Syntax: XOR( { Logical L }+ )

Constraints: Shall have 1 or more parameters.

Semantics: Computes the logical XOR of the parameters such that the result is an addition modulo 2. If an even number of parameters is TRUE it returns FALSE, if an odd number of parameters is TRUE it returns TRUE. When given one parameter, this has the effect of converting that one parameter into a Logical value.

6.16 Mathematical Functions

6.16.1 General

This section describes functions for various mathematical functions, including trigonometric functions like SIN 6.16.55). Note that the constraint text presumes that a value of type Number is a real number (no imaginary component). Unless noted otherwise, all angle measurements are in radians.

6.16.2 ABS

Summary: Return the absolute (nonnegative) value.

Syntax: ABS( Number N )

Semantics: If N < 0, returns -N, otherwise returns N.

6.16.3 ACOS

Summary: Returns the principal value of the arc cosine of a number. The angle is returned in radians.

Syntax: ACOS( Number N )

Constraints: -1.0 ≤ N ≤ 1.0.

Semantics: Computes the arc cosine of a number, in radians.

ACOS (N) = \frac{π}{2} - [N + \frac{1}{2 \cdot 3} N^{3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5} N^{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7} N^{7} + ...]

Returns a principal value 0 ≤ result ≤ π.

See also COS 6.16.19, RADIANS 6.16.49, DEGREES 6.16.25

6.16.4 ACOSH

Summary: Return the principal value of the inverse hyperbolic cosine.

Syntax: ACOSH( Number N )

Semantics: Computes the principal value of the inverse hyperbolic cosine.

Constraints: N ≥ 1

ACOSH (N) = \ln (N + \sqrt{N^{2} - 1})

6.16.5 ACOT

Summary: Return the principal value of the arc cotangent of a number. The angle is returned in radians.

Syntax: ACOT( Number N )

Semantics: Computes the arc cotangent of a number, in radians.

Returns a principal value 0 < result < π.

See also COT 6.16.21, ATAN 6.16.9, TAN 6.16.69, RADIANS 6.16.49, DEGREES 6.16.25

6.16.6 ACOTH

Summary: Return the hyperbolic arc cotangent

Syntax: ACOTH( Number N )

Semantics: Computes the hyperbolic arc cotangent. The hyperbolic arc cotangent is an analog of the ordinary (circular) arc cotangent.

Constraints: ABS(N) > 1

ACOTH (N) = \frac{1}{2} \ln (\frac{x + 1}{x - 1})

6.16.7 ASIN

Summary: Return the principal value of the arc sine of a number. The angle is returned in radians.

Syntax: ASIN( Number N )

Semantics: Computes the arc sine of a number, in radians.

Constraints: -1 ≤ N ≤ 1.

ASIN (N) = N + \frac{1}{2 \cdot 3} N^{3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5} N^{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7} N^{7} + .. .

Returns a principal value -π/2 ≤ result ≤ π/2.

See also SIN 6.16.55, RADIANS 6.16.49, DEGREES 6.16.25

6.16.8 ASINH

Summary: Return the principal value of the inverse hyperbolic sine

Syntax: ASINH( Number N )

Semantics: Computes the principal value of the inverse hyperbolic sine.

ASINH (N) = \ln (N + \sqrt{N^{2} + 1})

6.16.9 ATAN

Summary: Return the principal value of the arc tangent of a number. The angle is returned in radians.

Syntax: ATAN( Number N )

Semantics: Computes the arc tangent of a number, in radians.

Returns a principal value -π/2 < result < π/2.

See also ATAN2 6.16.10, TAN 6.16.69, RADIANS 6.16.49, DEGREES 6.16.25

6.16.10 ATAN2

Summary: Returns the principal value of the arc tangent given a coordinate of two numbers.

The angle is returned in radians.

Syntax: ATAN2( Number x ; Number y )

Constraints: x ≠ 0 or y ≠ 0

Semantics: Computes the arc tangent of two numbers (the x and y coordinates of a point), in radians. This is similar to ATAN(y/x), but the signs of the two numbers are taken into account so that the result covers the full range from -π to +π. ATAN2(0;0) is implementation-defined, evaluators may return 0 or an Error.

Returns a principal value -π < result ≤ π.

See also ATAN 6.16.9, TAN 6.16.69, RADIANS 6.16.49, DEGREES 6.16.25

6.16.11 ATANH

Summary: Return the principal value of the inverse hyperbolic tangent

Syntax: ATANH( Number N )

Semantics: Computes the principal value of the inverse hyperbolic tangent.

Constraints: -1 < N < 1

ATANH (N) = \frac{1}{2} \ln (\frac{1 + N}{1 - N})

See also COSH 6.16.20, SINH 6.16.56, ASINH 6.16.8, ACOSH 6.16.4, ATAN 6.16.9, ATAN2 6.16.10, FISHER 6.18.26

6.16.12 BESSELI

Summary: Returns the modified Bessel function of integer order In(X).

Syntax: BESSELI( Number X ; Number N )

Constraints: N ≥ 0, INT(N) = N; Evaluators may evaluate expressions where N ≥ 0 returns a non-error value.

Semantics: Computes the modified Bessel function of integer order In(X). N is also known as the order.

See also BESSELJ 6.16.13, BESSELK 6.16.14, BESSELY 6.16.15, INT 6.17.2

6.16.13 BESSELJ

Summary: Returns the Bessel function of integer order Jn(X) (cylinder function)

Syntax: BESSELJ( Number X ; Number N )

Constraints: N ≥ 0, INT(N) = N; Evaluators may evaluate expressions where N ≥ 0 returns a non-error value.

Semantics: Computes the Bessel function of integer order Jn(X). N is also known as the order.

See also BESSELI 6.16.12, BESSELK 6.16.14, BESSELY 6.16.15, INT 6.17.2

6.16.14 BESSELK

Summary: Returns the modified Bessel function of integer order Kn(x).

Syntax: BESSELK( Number X ; Number N )

Constraints: X ≠ 0, N ≥ 0, INT(N) = N; Evaluators may evaluate expressions where N ≥ 0 returns a non-error value.

Semantics: Computes the Bessel function of integer order Kn(x). N is also known as the order.

See also BESSELI 6.16.12, BESSELJ 6.16.13, BESSELY 6.16.15, INT 6.17.2

6.16.15 BESSELY

Summary: Returns the Bessel function of integer order Yn(X), also known as the Neumann function.

Syntax: BESSELY( Number X ; Number N )

Constraints: X ≠ 0, N ≥ 0, INT(N) = N; Evaluators may evaluate expressions where N ≥ 0 returns a non-error value.

Semantics: Computes Bessel function of integer order Yn(X), also known as the Neumann function. N is also known as the order.

See also BESSELI 6.16.12, BESSELJ 6.16.13, BESSELK 6.16.14, INT 6.17.2

6.16.16 COMBIN

Summary: Returns the number of different R-length sets that can be selected from N items.

Syntax: COMBIN( Integer N ; Integer R )

Constraints: N ≥ 0, R ≥ 0, R ≤ N

Semantics: COMBIN returns the binomial coefficient, which is the number of different R-length sets that can be selected from N items. Since they are sets, order in the sets is not relevant. The parameters are truncated (using INT) before use. For example, if a jar contains five marbles, each one a distinct color, the number of different three-marble groups COMBIN(5;3) = 10. The result is

(\begin{matrix} N \\ R \end{matrix}) = \frac{PERMUT}{R!} = \frac{N!}{R! (N - R)!}

Note that if order is important, use PERMUT instead.

6.16.17 COMBINA

Summary: Returns the number of combinations with repetitions.

Syntax: COMBINA( Integer N ; Integer M )

Constraints: N ≥ 0, M ≥ 0, N ≥ M; Evaluators may evaluate expressions where N ≥ 0, M ≥ 0 returns a non-error value.

Semantics: Returns the number of possible combinations of M objects out of N possible ones, with repetitions allowed. Actual arguments that are not integers are truncated (using INT) before use. The result is

(\begin{matrix} N + M - 1 \\ N - 1 \end{matrix})

6.16.18 CONVERT

Summary: Returns a number converted from one unit system into another.

Syntax: CONVERT( Number N ; Text From ; Text Into )

Constraints: From and Into shall be legal units, and shall be in the same unit group.

Semantics: Returns the number converted from the unit identified by From into the unit identified by Into. A unit is a unit symbol , optionally preceded by a unit prefix (either a decimal prefix or a binary prefix, as specified in Table 25 - Decimal Prefixes for use in CONVERT and Table 26 - Binary prefixes for use in CONVERT respectively). Units (including both the unit symbol and the optional unit prefix) are case-sensitive.

Evaluators claiming to implement this function shall support at least the following unit symbols (with conversions between them and other units in the same group):

Table 24 - Unit names

Unit group

Unit symbol

Description

Area

"uk_acre"	International acre (using international feet), exactly 4046.8564224 m2; normally not used for U.S. land areas
"us_acre"	U.S. survey/statute acre (using U.S. survey/statute feet), exactly 4046+13525426/15499969 m 2
"ang2" or "ang^2" *	Square angstrom (an Angstrom is exactly 10-10 m)
"ar" *	are, 100 m2 (not abbreviated as “a”)
"ft2" or "ft^2"	Square international feet (1 foot is exactly 0.3048 m)
"ha"	hectare, 10000 m2
"in2" or "in^2"	Square international inches (1 inch is exactly 2.54 cm)
"ly2" or "ly^2"	Square light-year (where year=365.25 days)
"m2" or "m^2" *	Square meters
"Morgen"	Morgen, 2500 m2
"mi2" or "mi^2"	Square international miles
"Nmi2" or "Nmi^2"	Square nautical miles (1 nautical mile is 1852 m)
"Pica2" or "Pica^2" or "picapt2" or "picapt^2"	Square Pica Point (one Pica point is 1/72 inch)
"pica2" or "pica^2"	Square Pica (one Pica is 1/6 inch)
"yd2" or "yd^2"	Square international yards (1 yard is 0.9144 m)

Distance (Length)

"ang" *	Angstrom, exactly 10-10 m
"ell"	Ell, exactly 45 international inches
"ft"	International Foot, exactly 0.3048 m and also exactly 12 international inches.
"in"	International Inch, exactly 2.54 cm.
"ly" *	Light-year, (299792458 m/s) (3600 s/hr) (24 hr/day) (365.25 day)
"m" *	Meter
"mi"	International Mile, exactly 1609.344 m and exactly 5280 international feet. This is not a U.S. survey/statute mile (see “survey_mi”) nor a nautical mile (see “Nmi”), but this is the mile normally used in the U.S. customary system
"Nmi"	International nautical mile, exactly 1852 m. Note that this is not the obsolete U.S. nautical mile nor the Admiralty mile.
"parsec" or "pc" *	Distance from sun to a point having heliocentric parallax of one second (used for stellar distance), AU/tan(1/3600 degree) where an AU is exactly 149,597,870.691 kilometers. *
"Pica" or "picapt"	Pica point (1/72 inch)
"pica"	Pica (1/6 inch)
"survey_mi"	U.S. survey "mile, aka U.S. statute mile, exactly 6336000/3937 m; used in some U.S. maps. This is not the mile (see “mi”) normally used in the U.S.
"yd"	International yard, exactly 0.9144 m and exactly 3 international feet.

Energy

"BTU" ("btu")	International Table British Thermal Unit
"c" *	Thermodynamic calorie, 4.184 J. This is not a dietary Calorie (kilocalorie). For high accuracy, use Joule, due to the many conflicting definitions of calorie.
"cal" *	International Table (IT) calorie, 4.1868 J. This is not a dietary Calorie (kilocalorie). For high accuracy, use Joule, due to the many conflicting definitions of calorie.
"e" *	Erg
"eV" ("ev") *	Electron volt (eV preferred)
"flb"	Foot-pound (international foot, avoirdupois pound)
"HPh" ("hh")	Horsepower-hour (HPh preferred)
"J" *	Joule
"Wh" ("wh") *	Watt-hour

Force (Weight)

"dyn" ("dy") *	Dyne
"N" *	Newton
"lbf"	Pound force (see “lbm” for pound mass)
"pond" *	Pond, gravitational force on a mass of one gram, 9.80665E-3 N.

Information

"bit" * †	bit
"byte" * †	byte = 8 bits

Magnetic Flux Density

"ga" *	Gauss
"T" *	Tesla

Mass

"g" *	Gram
"grain"	Grain, 1/7000 international pound mass (avoirdupois) (U.S. usage).
"cwt" ("shweight")	U.S. (short) hundredweight, 100 lbm
"uk_cwt" or "lcwt" ("hweight")	Imperial hundredweight, aka long hundredweight; 112 lbm
"lbm"	International pound mass (avoirdupois), exactly 453.59237 g (see “lbf” for pound force)
"stone"	14 international pound mass (avoirdupois)
"ton"	2000 international pound mass (avoirdupois) (U.S. usage). Note that there are many other measures also called “ton”; in particular, this is not a metric ton (tonne).
"ozm"	Ounce mass (avoirdupois), exactly 1/16 of an international pound mass (avoirdupois) (see “oz” for fluid ounce)
"sg"	Slug; 32.174 international pound mass (avoirdupois)
"u" *	U (atomic mass unit)
"uk_ton" or "LTON" ("brton")	Imperial ton, aka “long ton”, "deadweight ton", or "weight ton". 2240 lbm.

Power

"HP" ("h")	Mechanical horsepower aka Imperial horsepower. 550 foot-pounds per second. The unit “h” is deprecated and should be replaced with “HP”.
"PS"	Pferdestärke (German “horse strength”, close but not identical to “HP”), the amount of power to lift a mass of 75 kilograms in one second against the earth gravitation between a distance of one meter, approximately 735.49875 W.
"W" ("w") *	Watt

Pressure

"atm" ("at") *	Atmosphere
"mmHg" *	mm of Mercury
"Pa" *	Pascal Note: “P” or “p” in user input as abbreviations for Pascal may be accepted by implementations, but should be stored as “Pa”..
"psi"	Pounds per square inch, using avoirdupois pounds and international inches.
"Torr"	Torr, exactly 101325/760 Pa (this is close but not equal to mmHg)

Speed

"admkn"	Admiralty knot, exactly 6080 international feet/hour.
"kn"	Knot, exactly one Nautical mile per hour or exactly 1852/3600 m/s. Note that this is not an Admiralty knot (“admkn”).
"m/h" or "m/hr" *	Meters per hour
"m/s" or "m/sec" *	Meters per second
"mph"	Miles per hour (international miles)

Temperature

"C" ("cel")	degrees Celsius
"F" ("fah")	degrees Fahrenheit
"K" ("kel") *	Kelvin
"Rank"	degrees Rankine
"Reau"	degrees Réaumur; °C = °Ré · 5/4.

Time

"day" or "d"	Day (exactly 24 hours)
"hr"	Hour (exactly 60 minutes)
"mn" or "min"	Minute (exactly 60 seconds)
"sec" or "s" *	Second (“s” is the official abbreviation of this SI base unit, while “sec” is a widely-recognized abbreviation in the CONVERT function) *
"yr"	Year (exactly 365.25 days, for purposes of this function)

Volume

"ang3" or "ang^3" *	Cubic angstrom
"barrel"	U.S. oil barrel, exactly 42 U.S. customary gallons (liquid). Note that many other units are also called barrels (e.g., a beer barrel in the U.K. is 36 Imperial gallons)
"bushel"	U.S. bushel (not Imperial bushel), interpreted as volume
"cup"	Cup (U.S. customary liquid measure)
"ft3" or "ft^3"	Cubic international feet
"gal"	Gallon (U.S. customary liquid measure), 3.785411784 liters.
"GRT" ("regton")	Gross Registered Ton, 100 cubic (international) feet
"in3" or "in^3"	Cubic international inch
"l" or "L" ("lt") *	Liter
"ly3" or "ly^3"	Cubic light-year
"m3" or "m^3" *	Cubic meter
"mi3" or "mi^3"	Cubic international mile
"MTON"	Measurement ton aka “freight ton”, 40 cubic feet
"Nmi3" or "Nmi^3"	Cubic nautical mile
"oz"	Fluid ounce (U.S. customary liquid measure; see “ozm” for ounce mass)
"Pica3" or "Pica^3" "picapt3" or "picapt^3"	Cubic Pica Point (one Pica point is 1/72 inch)
"pica3" or "pica^3"	Cubic Pica (one Pica is 1/6 inch)
"pt" or "us_pt"	U.S. Pint (liquid measure)
"qt"	Quart (U.S. customary liquid measure). This is 0.946352946 liters, and thus not the same as the U.S. dry quart (1.101220 liters), nor is this the same as the Imperial quart (as used in the U.K. and Canada, which is 1.1365225 liters exactly)
"tbs"	Tablespoon (U.S. customary, traditional meaning). This shall be 0.5 U.S. fluid ounce, not 15mL (common in U.S.) or 20mL (common in Australia).
"tsp"	Teaspoon (U.S. customary, traditional meaning), 1/6 fluid ounce in U.S. customary measure. This is not the 1/8 Imperial fl. oz. per Imperial units nor the modern teaspoon of 5 mL currently used in the U.S.; see “tspm”
"tspm"	Modern teaspoon, 5mL
"uk_gal"	U.K. / Imperial gallon, 4.54609 liters.
"uk_pt"	U.K. / Imperial pint,1/8 of a UK gallon.
"uk_qt"	U.K. / Imperial quart,1/4 of a UK gallon.
"yd3" or "yd^3"	Cubic international yard

If a conversion factor (as listed above) is not exact, an implementation may use a more accurate conversion factor instead.

Implementation-defined unit names should contain a 'FULL STOP' (U+002E) character.

Evaluators shall support decimal prefixes for unit symbols marked with * and binary prefixes for unit symbols marked with †. Evaluators should not support prefixes for other unit symbols.

The unit symbols in parentheses are deprecated unit symbols; evaluators shall support these unit symbols.

Evaluators should use internationally-standardized unit name abbreviations for such additions where possible. Evaluators may support the obsolete symbols “p” and “P” as unit names for Pascals.

For purposes of this function, a year is exactly 365.25 days long.

Evaluators claiming to support this function shall permit the unit decimal prefixes specified in Table 25 - Decimal Prefixes for use in CONVERT to be prepended to any unit symbol marked with * in Table 24 - Unit names. Adding a unit prefix indicates multiplication of the (scalar) unit by the given prefix value; for example km indicates kilometres, and km2 or km^2 indicate square kilometres.

Table 25 - Decimal Prefixes for use in CONVERT

Unit Prefix	Description	Prefix Value
"Y"	yotta	1E+24
"Z"	zetta	1E+21
"E"	exa	1E+18
"P"	peta	1E+15
"T"	tera	1E+12
"G"	giga	1E+09
"M"	mega	1E+06
"k"	kilo	1E+03
"h"	hecto	1E+02
"da" or "e"	deka ( Note: “e” is not a standard SI prefix	1E+01
"d"	deci	1E-01
"c"	centi	1E-02
"m"	milli	1E-03
"u"	micro Note: this is “u”, not the standard SI µ	1E-06
"n"	nano	1E-09
"p"	pico	1E-12
"f"	femto	1E-15
"a"	atto	1E-18
"z"	zepto	1E-21
"y"	yocto	1E-24

Note: The prefix “e” for 10 1 is nonstandard and included for backward compatibility with legacy applications and documents.

The unit names marked with † in Table 24 - Unit names (see the Information Unit group) shall also support the following binary prefixes per IEC 60027-2:

Table 26 - Binary prefixes for use in CONVERT

Binary Unit Prefix	Description	Prefix Value	Derived from
"Yi"	yobi	2^80 = 1 208 925 819 614 629 174 706 176	yotta
"Zi"	zebi	2^70 = 1 180 591 620 717 411 303 424	zetta
"Ei"	exbi	2^60 = 1 152 921 504 606 846 976	exa
"Pi"	pebi	2^50 = 1 125 899 906 842 624	peta
"Ti"	tebi	2^40 = 1 099 511 627 776	tera
"Gi"	gibi	2^30 = 1 073 741 824	giga
"Mi"	mebi	2^20 = 1 048 576	mega
"Ki"	kibi	2^10 = 1024	kilo

In the case where there is a naming conflict (a unit name with a prefix is the same as an unprefixed name), the unprefixed name shall take precedence.

Evaluators may implement this conversion by first converting to some SI unit (e.g., meter and kilogram), and then convert again to the final unit.

6.16.19 COS

Summary: Return the cosine of an angle specified in radians.

Syntax: COS( Number N )

Semantics: Computes the cosine of an angle specified in radians.

COS (N) = 1 - \frac{N^{2}}{2!} + \frac{N^{4}}{4!} - \frac{N^{6}}{6!} + .. .

See also ACOS 6.16.3, RADIANS 6.16.49, DEGREES 6.16.25

6.16.20 COSH

Summary: Return the hyperbolic cosine of the given hyperbolic angle.

Syntax: COSH( Number N )

Semantics: Computes the hyperbolic cosine of a hyperbolic angle. The hyperbolic cosine is an analog of the ordinary (circular) cosine. The points (cosh t, sinh t) define the right half of the equilateral hyperbola, just as the points (cos t, sin t) define the points of a circle.

COSH (N) = \frac{e^{N} + e^{- N}}{2}

See also ACOSH 6.16.4, SINH 6.16.56, TANH 6.16.70

6.16.21 COT

Summary: Return the cotangent of an angle specified in radians.

Syntax: COT( Number N )

Semantics: Computes the cotangent of an angle specified in radians.

COT(x) = 1 / TAN(x)

See also ACOT 6.16.5, TAN 6.16.69, RADIANS 6.16.49, DEGREES 6.16.25, SIN 6.16.55, COS 6.16.19

6.16.22 COTH

Summary: Return the hyperbolic cotangent of the given hyperbolic angle.

Syntax: COTH( Number N )

Semantics: Computes the hyperbolic cotangent of a hyperbolic angle. The hyperbolic cotangent is an analog of the ordinary (circular) cotangent.

Constraints: N ≠ 0

\coth (N) = \frac{1}{\tanh (N)} = \frac{\cosh (N)}{\sinh (N)} = \frac{e^{N} + e^{- N}}{e^{N} - e^{- N}}

See also ACOSH 6.16.4, COSH 6.16.20, SINH 6.16.56, TANH 6.16.70

6.16.23 CSC

Summary: Return the cosecant of an angle specified in radians.

Syntax: CSC( Number N )

Semantics: Computes the cosecant cosine of an angle specified in radians. Equivalent to:

1 / SIN(N)

6.16.24 CSCH

Summary: Return the hyperbolic cosecant of the given angle specified in radians.

Syntax: CSCH( Number N )

Semantics: Computes the hyperbolic cosecant of a hyperbolic angle. This is equivalent to:

1 / SINH(N)

6.16.25 DEGREES

Summary: Convert radians to degrees.

Syntax: DEGREES( Number N )

Semantics: Converts a number in radians into a number in degrees. DEGREES(N) is equal to N * 180 / π.

6.16.26 DELTA

Summary: Report if two numbers are equal, returns 1 if they are equal.

Syntax: DELTA( Number X [ ; Number Y = 0 ] )

Semantics: If X and Y are equal, return 1, else 0. Y is set to 0 if omitted.

6.16.27 ERF

Summary: Calculates the error function.

Syntax: ERF( Number Z0 [ ; Number Z1 ] )

Semantics: With a single argument, returns the error function of Z0:

ERF (Z 0) = \frac{2}{\sqrt{(π)}} \int_{0}^{Z 0} e^{- t^{2}} dt

With two arguments, returns

ERF (Z 0; Z 1) = \frac{2}{\sqrt{(π)}} \int_{Z 0}^{Z 1} e^{- t^{2}} dt

6.16.28 ERFC

Summary: Calculates the complementary error function.

Syntax: ERFC( Number Z )

Semantics: returns the complementary error function of Z: ERFC(Z) = 1 – ERF(Z)

6.16.29 EUROCONVERT

Summary: Converts a Number, representing a value in one European currency, to an equivalent value in another European currency, according to the fixed conversion rates defined by the Council of the European Union.

Syntax: EUROCONVERT( Number N ; Text From ; Text To [ ; Logical FullPrecision = FALSE [ ; Integer TriangulationPrecision ] ] )

Constraints: From and To shall be known to the evaluator. TriangulationPrecision shall be ≥ 3, if not omitted.

If an evaluator does not support the parameters FullPrecision and TriangulationPrecision, FullPrecision should be assumed to be false.

Semantics: Returns the given money value of a conversion from From currency into To currency. Both From and To shall be the official [ISO4217] abbreviation for the given currency; note that these are in upper case, but the function accepts lower case or mixed case as well. If From and To are equal currencies, the value N is returned, no precision or triangulation is applied.

The function shall use the rates of exchange as set by the European Commission, as follows:

Table 27 - EUROCONVERT

From	To	Rate	Currency	Decimals
"EUR"	"ATS"	13.7603	Austrian Schilling	2
"EUR"	"BEF"	40.3399	Belgian Franc	0
"EUR"	"DEM"	1.95583	German Mark	2
"EUR"	"ESP"	166.386	Spanish Peseta	0
"EUR"	"FIM"	5.94573	Finnish Markka	2
"EUR"	"FRF"	6.55957	French Franc	2
"EUR"	"IEP"	0.787564	Irish Pound	2
"EUR"	"ITL"	1936.27	Italian Lira	0
"EUR"	"LUF"	40.3399	Luxembourg Franc	0
"EUR"	"NLG"	2.20371	Dutch Guilder	2
"EUR"	"PTE"	200.482	Portuguese Escudo	2
"EUR"	"GRD"	340.750	Greek Drachma	2
"EUR"	"SIT"	239.640	Slovenian Tolar	2
“EUR”	“MTL”	0.429300	Maltese Lira	2
“EUR”	“CYP”	0.585274	Cypriot Pound	2
"EUR"	"SKK"	30.1260	Slovak Koruna	2

As new member countries adopt the Euro, new conversion rates will become active and evaluators may add them using the respective [ISO4217] codes and fixed rates as defined by the European Council, on the basis of a European Commission proposal.

Note:
The European Commission's Euro entry page is http://ec.europa.eu/euro/
The conversion rates and triangulation rules are available at http://ec.europa.eu/economy_finance/euro/adoption/conversion/index_en.htm with links to the European Council Regulation legal documents at the http://eur-lex.europa.eu/ European Union law database server.

If FullPrecision is omitted or FALSE, the result is rounded according to the decimals of the To currency. If FullPrecision is TRUE the result is not rounded.

If TriangulationPrecision is given and ≥ 3, the intermediate result of a triangular conversion (currency1,EUR,currency2) is rounded to that precision. If TriangulationPrecision is omitted, the intermediate result is not rounded. Also if To currency is “EUR”, TriangulationPrecision precision is used as if triangulation was needed and conversion from EUR to EUR was applied.

6.16.30 EVEN

Summary: Rounds a number up to the nearest even integer. Rounding is away from zero.

Syntax: EVEN( Number N )

Semantics: Returns the even integer whose sign is the same as N's and whose absolute value is greater than or equal to the absolute value of N.

6.16.31 EXP

Summary: Returns e raised by the given number.

Syntax: EXP( Number X )

Summary: Return factorial (!).

Semantics: Computes

e^{X} = 1 + \frac{X}{1!} + \frac{X^{2}}{2!} + \frac{X^{3}}{3!} + \frac{X^{n}}{n!} + .. .

6.16.32 FACT

Syntax: FACT( Integer F )

Semantics: Return the factorial

Constraints: F ≥ 0

F! = F \cdot (F - 1) \cdot (F - 2) \cdot ... 1

F(0) = F(1) = 1.

6.16.33 FACTDOUBLE

Summary: Returns double factorial (!!).

Syntax: FACTDOUBLE( Integer F )

Double factorial is computed by multiplying every other number in the 1..N range, with N always being included.

Constraints: F ≥ 0

Semantics: Return

F! = F \cdot (F - 2) \cdot (F - 4) \cdot ... 1

6.16.34 GAMMA

Summary: Return gamma function value.

Syntax: GAMMA( Number N )

Constraints: N ≠ 0 and N not a negative integer.

Semantics: Return

Γ(N) = \int_{0}^{\infty} t^{N - 1} e^{- t} dt

with Γ(N + 1) = N * Γ(N). Note that for non-negative integers N, Γ(N + 1) = N! = FACT(N). Note that GAMMA can accept non-integers.

6.16.35 GAMMALN

Summary: Returns the natural logarithm of the GAMMA function.

Syntax: GAMMALN( Number X )

Constraints: For each X, X > 0

Semantics: Returns the same value as LN(GAMMA(X))

6.16.36 GCD

Summary: Returns the greatest common divisor (GCD)

Syntax: GCD( { NumberSequenceList X }+ )

Constraints: For all a in X: INT(a) ≥ 0 and for at least one a in X: INT(a) > 0

Semantics: Return the largest integer N such that for every a in X: INT(a) is a multiple of N.

Note: If for all a in X: INT(a) = 0 the return value is implementation-defined but is either an Error or 0.

6.16.37 GESTEP

Summary: Returns 1 if a number is greater than or equal to another number, else returns 0.

Syntax: GESTEP( Number X [ ; Number Step = 0 ] )

Semantics: Number X is tested against number Step. If greater or equal 1 is returned, else 0. The second parameter is assumed 0 if omitted. If one of the parameters is not a Number, the function results in an Error.

6.16.38 LCM

Summary: Returns the least common multiplier

Syntax: LCM( { NumberSequenceList X }+ )

Constraints: For all in X: INT(X) = X, X ≥ 0

Semantics: Return the smallest integer that is the multiple of the given values. Each value has INT applied to it first. Note that if given two numbers, ABS(a * b) = LCM(a;b) * GCD(a;b).

6.16.39 LN

Summary: Return the natural logarithm of a number.

Syntax: LN( Number X )

Semantics: Computes the natural logarithm (base e) of the given number.

Constraints: X > 0

LN (X) = 2 [\frac{x - 1}{x + 1} + \frac{1}{3} {(\frac{x - 1}{x + 1})}^{3} + \frac{1}{5} {(\frac{x - 1}{x + 1})}^{5} + ...]

See also LOG 6.16.40, LOG10 6.16.41, POWER 6.16.46, EXP 6.16.31

6.16.40 LOG

Summary: Return the logarithm of a number in a specified base.

Syntax: LOG( Number N [ ; Number Base = 10 ] )

Semantics: Computes the logarithm of a number in the specified base. Note that if the base is not specified, the logarithm base 10 is returned.

Constraints: N > 0

See also LOG10 6.16.41, LN 6.16.39, POWER 6.16.46, EXP 6.16.31

6.16.41 LOG10

Summary: Return the base 10 logarithm of a number.

Syntax: LOG10( Number N )

Semantics: Computes the base 10 logarithm of a number.

Constraints: N > 0

See also LOG 6.16.40, LN 6.16.39, POWER 6.16.46, EXP 6.16.31

6.16.42 MOD

Summary: Return the remainder when one number is divided by another number.

Syntax: MOD( Number A ; Number B )

Semantics: Computes the remainder of A / B. The remainder has the same sign as B.

Constraints: B != 0

6.16.43 MULTINOMIAL

Summary: Returns the multinomial for the given values.

Syntax: MULTINOMIAL( { NumberSequence A }+ )

Semantics: Returns the multinomial of the sequence A = (a1, a2, ..., an). Multinomial is defined as FACT(a1 + a2 +...+ an) / (FACT(a1) * FACT(a2) *...* FACT(an))

6.16.44 ODD

Summary: Rounds a number up to the nearest odd integer, where "up" means "away from 0".

Syntax: ODD( Number N )

Semantics: Returns the odd integer whose sign is the same as N's and whose absolute value is greater than or equal to the absolute value of N. In other words, any "rounding" is away from zero. By definition, ODD(0) is 1.

6.16.45 PI

Summary: Return the approximate value of π.

Syntax: PI()

Semantics: This function takes no arguments and returns the (approximate) value of π (pi). Evaluators should use the closest possible numerical representation that is possible in their representation of numbers.

6.16.46 POWER

Summary: Return the value of one number raised to the power of another number.

Syntax: POWER( Number A ; Number B )

Semantics: Computes A raised to the power B.

•POWER(0,0) is implementation-defined, but shall be one of 0,1, or an Error.
•POWER(0,B), where B < 0, shall return an Error.
•POWER(A,B), where A ≤ 0 and INT(B) != B, is implementation-defined.

See also LOG 6.16.40, LOG10 6.16.41, LN 6.16.39, EXP 6.16.31

6.16.47 PRODUCT

Summary: Multiply the set of numbers, including all numbers inside ranges.

Syntax: PRODUCT( { NumberSequenceList N }+ )

Semantics: Returns the product of the Numbers (and only the Numbers, i.e., not Text inside ranges).

6.16.48 QUOTIENT

Summary: Return the integer portion of a division.

Syntax: QUOTIENT( Number A ; Number B )

Semantics: Return the integer portion of a division.

Constraints: B ≠ 0

6.16.49 RADIANS

Summary: Convert degrees to radians.

Syntax: RADIANS( Number N )

Semantics: Converts a number in degrees into a number in radians. RADIANS(N) is equal to N * PI() / 180.

6.16.50 RAND

Summary: Return a random number between 0 (inclusive) and 1 (exclusive).

Syntax: RAND()

Semantics: This function takes no arguments and returns a random number between 0 (inclusive) and 1 (exclusive). Note that unlike most functions, this function will typically return different values when called each time with the same (empty set of) parameters.

6.16.51 RANDBETWEEN

Summary: Return a random integer number between A and B.

Syntax: RANDBETWEEN( Integer A ; Integer B )

Returns: Integer

Constraints: A ≤ B

Semantics: The function returns a random integer number between A and B inclusive. Note that unlike most functions, this function will often return different values when called each time with the same parameters.

6.16.52 SEC

Summary: Return the secant of an angle specified in radians.

Syntax: SEC( Number N )

Semantics: Computes the secant cosine of an angle specified in radians. Equivalent to:

1 / COS(N)

6.16.53 SERIESSUM

Summary: Returns the sum of a power series.

Syntax: SERIESSUM( Number X ; Number N ; Number M ; Array Coefficients )

•X: the independent variable of the power series.
•N: the initial power to which X is to be raised.
•M: the increment by which to increase N for each term in the series.
•Coefficients: a set of coefficients by which each successive power of the variable X is multiplied.

All elements of Coefficients are of type Number.

X ≠ 0 if any of the exponents, which are generated from N and M, are negative.

Semantics: Returns a sum of powers of the number X.

With C being the number of coefficients the function is computed as:

SERIESSUM = \sum_{i = 1}^{C} {Coefficient}_{i} \cdot X^{(N + (i - 1) M)}

= 0 and all of the exponents are non-negative then

0^{0}

shall be set to 1 and

0^{(exponent > 0)}

shall be set to 0.

6.16.54 SIGN

Summary: Return the sign of a number.

Syntax: SIGN( Number N )

Semantics: If N < 0, returns -1; if N > 0, returns +1; if N = 0, returns 0.

6.16.55 SIN

Summary: Return the sine of an angle specified in radians.

Syntax: SIN( Number N )

Semantics: Computes the sine of an angle specified in radians.

SIN (N) = N - \frac{N^{3}}{3!} + \frac{N^{5}}{5!} - \frac{N^{7}}{7!} + .. .

See also ASIN 6.16.7, RADIANS 6.16.49, DEGREES 6.16.25

6.16.56 SINH

Summary: Return the hyperbolic sine of the given hyperbolic angle.

Syntax: SINH( Number N )

Semantics: Computes the hyperbolic sine of a hyperbolic angle. The hyperbolic sine is an analog of the ordinary (circular) sine. The points (cosh t, sinh t) define the right half of the equilateral hyperbola, just as the points (cos t, sin t) define the points of a circle.

SINH (N) = \frac{e^{N} - e^{- N}}{2}

See also ASINH 6.16.8, COSH 6.16.20, TANH 6.16.70

6.16.57 SECH

Summary: Return the hyperbolic secant of the given angle specified in radians.

Syntax: SECH( Number N )

Semantics: Computes the hyperbolic secant of a hyperbolic angle. This is equivalent to:

1 / COSH(N)

See also SINH 6.16.56, COSH 6.16.20, CSCH 6.16.24

6.16.58 SQRT

Summary: Return the square root of a number.

Syntax: SQRT( Number N )

Semantics: Returns the square root of a non-negative number. This function shall produce an Error if given a negative number; for producing complex numbers, see IMSQRT.

Constraints: N ≥ 0

See also POWER 6.16.46, IMSQRT 6.8.24, SQRTPI 6.16.59

6.16.59 SQRTPI

Summary: Return the square root of a number multiplied by π (pi).

Syntax: SQRTPI( Number N )

Semantics: Returns the square root of a non-negative number after it was first multiplied by π, that is, SQRT(N * PI()). This function shall produce an Error if given a negative number; for producing complex numbers, see IMSQRT.

Constraints: N ≥ 0

See also POWER 6.16.46, SQRT 6.16.58, PI 6.16.45, IMSQRT 6.8.24

6.16.60 SUBTOTAL

Summary: Evaluates a function on a range.

Syntax: SUBTOTAL( Integer Function ; NumberSequence Sequence )

Semantics: Computes a given function on a number sequence. The function is denoted by the first parameter: The difference from standard functions is that all members of the sequence are excluded which:

•include a call to SUBTOTAL in their formula
•are in a row that is hidden by a table:visibility=”filter” attribute of the <table:table-row> element (OpenDocument, Part 3, 19.754).
•are in a row that is hidden by a table:visibility=”collapse” attribute of the <table:table-row> element if the function ID is one of 101...111.

Table 28 - SUBTOTAL

Function	Exclude hidden by filter	Exclude hidden by filter or collapsed
AVERAGE	1	101
COUNT	2	102
COUNTA	3	103
MAX	4	104
MIN	5	105
PRODUCT	6	106
STDEV	7	107
STDEVP	8	108
SUM	9	109
VAR	10	110
VARP	11	111

6.16.61 SUM

Summary: Sum (add) the set of numbers, including all numbers in ranges.

Syntax: SUM( { NumberSequenceList N }+ )

Constraints: N != {}; Evaluators may evaluate expressions that do not meet this constraint.

Semantics: Adds Numbers (and only Numbers) together (see the text on conversions).

6.16.62 SUMIF

Summary: Sum the values of cells in a range that meet a criteria.

Syntax: SUMIF( ReferenceList|Reference R ; Criterion C [ ; Reference S ] )

Constraints: Does not accept constant values as the range parameter.

Semantics: Sums the values of type Number in the range R or S that meet the Criterion C (4.11.8).

If S is not given, R may be a reference list. If S is given, R shall not be a reference list with more than 1 references and an Error be generated if it was.

If the optional range S is included, then the values of S starting from the top left cell and matching the geometry of R (same number of rows and columns) are summed if the corresponding value in R meets the Criterion. The actual range S is not considered. If the resulting range exceeds the sheet bounds, column numbers larger than the maximum column and row numbers larger than the maximum row are silently ignored, no Error is generated for this case.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also COUNTIF 6.13.9, SUM 6.16.61, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.16.63 SUMIFS

Summary: Sum the values of cells in a range that meet multiple criteria in multiple ranges.

Syntax: SUMIFS( Reference R ; Reference R1 ; Criterion C1 [ ; Reference R2 ; Criterion C2 ]... )

Constraints: Does not accept constant values as the reference parameter.

Semantics: Sums the value of cells in range R that meet the Criterion C1 in the reference range R1 and the Criterion C2 in the reference range R2, and so on (4.11.8). All reference ranges shall have the same dimension and size, else an Error is returned. A logical AND is applied between each array result of each selection; an entry is counted only if the same position in each array is the result of a criteria match.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also AVERAGEIFS 6.18.6, COUNTIFS 6.13.10, SUMIF 6.16.62, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.16.64 SUMPRODUCT

Summary: Returns the sum of the products of the matrix elements.

Syntax: SUMPRODUCT( { ForceArray Array A }+ )

Constraints: All matrices shall have the same dimensions.

Semantics: Multiplies the corresponding elements of all matrices and returns the sum of them.

SUMPRODUCT (A_{1}, A_{2}, ..., A_{K}) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} (\prod_{k = 1}^{K} a_{k, mn})

denotes an element of the matrix

a_{k, mn}

A_{K}

6.16.65 SUMSQ

Summary: Sum (add) the set of squares of numbers, including all numbers in ranges

Syntax: SUMSQ( { NumberSequence N }+ )

Constraints: N != {}; Evaluators may evaluate expressions that do not meet this constraint.

Semantics: Adds squares of Numbers (and only Numbers) together. See the text on conversions.

6.16.66 SUMX2MY2

Summary: Returns the sum of the difference between the squares of the matrices A and B.

Syntax: SUMX2MY2( ForceArray Array A ; ForceArray Array B )

Constraints: Both matrices shall have the same dimensions.

Semantics: Sums up the differences of the corresponding elements squares for two matrices.

SUMX 2 MY 2 (A, B) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} (a_{mn}^{2} - b_{mn}^{2})

6.16.67 SUMX2PY2

Summary: Returns the total sum of the squares of the matrices A and B.

Syntax: SUMX2PY2( ForceArray Array A ; ForceArray Array B )

Constraints: Both matrices shall have the same dimensions.

Semantics: Sums up the squares of each element of the two matrices.

SUMX 2 PY 2 (A, B) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} (a_{mn}^{2} + b_{mn}^{2})

6.16.68 SUMXMY2

Summary: Returns the sum of the squares of the differences between matrix A and B.

Syntax: SUMXMY2( ForceArray Array A ; ForceArray Array B )

Constraints: Both matrices shall have the same dimensions.

Semantics: Sums up the squares of the differences of the corresponding elements for two matrices.

SUMXMY 2 (A, B) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} {(a_{mn} - b_{mn})}^{2}

6.16.69 TAN

Summary: Return the tangent of an angle specified in radians

Syntax: TAN( Number N )

Semantics: Computes the tangent of an angle specified in radians.

TAN(x) = SIN(x) / COS(x)

See also ATAN 6.16.9, ATAN2 6.16.10, RADIANS 6.16.49, DEGREES 6.16.25, SIN 6.16.55, COS 6.16.19, COT 6.16.21

6.16.70 TANH

Summary: Return the hyperbolic tangent of the given hyperbolic angle

Syntax: TANH( Number N )

Semantics: Computes the hyperbolic tangent of a hyperbolic angle. The hyperbolic tangent is an analog of the ordinary (circular) tangent. The points (cosh t, sinh t) define the right half of the equilateral hyperbola, just as the points (cos t, sin t) define the points of a circle.

TANH (N) = \frac{SINH (N)}{COSH (N)} = \frac{e^{N} - e^{- N}}{e^{N} + e^{- N}}

See also ATANH 6.16.11, SINH 6.16.56, COSH 6.16.20, FISHERINV 6.18.27

6.17 Rounding Functions

6.17.1 CEILING

Summary: Round a number N up to the nearest multiple of the second parameter, significance.

Syntax: CEILING( Number N [ ; [ Number Significance ] [ ; Number Mode ] ] )

Constraints: Both N and Significance shall be numeric and have the same sign if not 0.

Semantics: Rounds a number up to a multiple of the second number. If Significance is omitted or an empty parameter (two consecutive ;; semicolons) it is assumed to be -1 if N is negative and +1 if N is non-negative, making the function act like the normal mathematical ceiling function if Mode is not given or zero. If Mode is given and not equal to zero, the absolute value of N is rounded away from zero to a multiple of the absolute value of Significance and then the sign applied . If Mode is omitted or zero, rounding is toward positive infinity; the number is rounded to the smallest multiple of significance that is equal-to or greater than N. If any of the two parameters N or Significance is zero, the result is zero.

Note: Many application user interfaces have a CEILING function with only two parameters, and somewhat different semantics than given here (e.g., they operate as if there was a non-zero Mode value). These CEILING functions are inconsistent with the standard mathematical definition of CEILING.

6.17.2 INT

Summary: Rounds a number down to the nearest integer.

Syntax: INT( Number N )

Semantics: Returns the nearest integer whose value is less than or equal to N. Rounding is towards negative infinity.

6.17.3 FLOOR

Summary: Round a number N down to the nearest multiple of the second parameter, significance.

Syntax: FLOOR( Number N [ ; [ Number Significance ] [ ; Number Mode ] ] )

Constraints: Both N and Significance shall be numeric and have the same sign.

Semantics: Rounds a number down to a multiple of the second number. If Significance is omitted or an empty parameter (two consecutive ;; semicolons) it is assumed to be -1 if N is negative and +1 if N is non-negative, making the function act like the normal mathematical floor function if Mode is not given or zero. If Mode is given and not equal to zero, the absolute value of N is rounded toward zero to a multiple of the absolute value of Significance and then the sign applied . Otherwise, it rounds toward negative infinity, and the result is the largest multiple of Significance that is less than or equal to N. If any of the two parameters N or Significance is zero, the result is zero.

Note: Many application user interfaces have a FLOOR function with only two parameters, and somewhat different semantics than given here (e.g., they operate as if there was a non-zero Mode value). These FLOOR functions are inconsistent with the standard mathematical definition of FLOOR.

6.17.4 MROUND

Summary: Rounds the number to given multiple.

Syntax: MROUND( Number A ; Number B )

Semantics: Returns the number X, for which the following holds: X/B = INT(X / B) (B divides X), and for any other Y with the same property, ABS(Y – A) ≥ ABS(X - A). In case that two such X exist, the greater one is the result. In less formal language, this function rounds the number A to multiples of B.

See also ABS 6.16.2, INT 6.17.2, ROUND 6.17.5

6.17.5 ROUND

Summary: Rounds the value X to the nearest multiple of the power of 10 specified by Digits.

Syntax: ROUND( Number X [ ; Number Digits = 0 ] )

Semantics: Round number X to the precision specified by Digits. The number X is rounded to the nearest power of 10 given by 10 −Digits. If Digits is zero, or absent, round to the nearest decimal integer. If Digits is non-negative, round to the specified number of decimal places. If Digits is negative, round to the left of the decimal point by -Digits places. If X is halfway between the two nearest values, the result shall round away from zero. Note that if X is a Number, and Digits ≤ 0, the results will always be an integer (without a fractional component).

6.17.6 ROUNDDOWN

Summary: Rounds the value X towards zero to the number of digits specified by Digits.

Syntax: ROUNDDOWN( Number X [ ; Integer Digits = 0 ] )

Semantics: Round X towards zero, to the precision specified by Digits. The number returned is a multiple of 10−Digits. If Digits is zero, or absent, round to the largest decimal integer whose absolute value is smaller or equal to the absolute value of X. If Digits is positive, round towards zero to the specified number of decimal places. If Digits is negative, round towards zero to the left of the decimal point by -Digits places.

See also TRUNC 6.17.8, INT 6.17.2, ROUND 6.17.5, ROUNDUP 6.17.7

6.17.7 ROUNDUP

Summary: Rounds the value X away from zero to the number of digits specified by Digits

Syntax: ROUNDUP( Number X [ ; Integer Digits = 0 ] )

Semantics: Round X away from zero, to the precision specified by Digits. The number returned is a multiple of 10−Digits. If Digits is zero, or absent, round to the smallest decimal integer whose absolute value is larger or equal to the absolute value of X. If Digits is positive, round away from zero to the specified number of decimal places. If Digits is negative, round away from zero to the left of the decimal point by -Digits places.

See also TRUNC 6.17.8, INT 6.17.2, ROUND 6.17.5, ROUNDDOWN 6.17.6

6.17.8 TRUNC

Summary: Truncate a number to a specified number of digits.

Syntax: TRUNC( Number A ; Integer B )

Semantics: Truncate number A to the number of digits specified by B. If B is zero, or absent, truncate to an integer. If B is positive, truncate to the specified number of decimal places. If B is negative, truncate to the left of the decimal point.

6.18 Statistical Functions

6.18.1 General

The following are statistical functions (functions that report information on a set of numbers). Some functions that could also be considered statistical functions, such as SUM, are listed elsewhere.

6.18.2 AVEDEV

Summary: Calculates the average of the absolute deviations of the values in list.

Syntax: AVEDEV( { NumberSequenceList N }+ )

Semantics: For a list N containing n numbers x1 to xn, with average x, AVEDEV(N) is equal to:

\frac{1}{n} \sum_{i = 1}^{n} | (x_{i} - \bar{x}) |

6.18.3 AVERAGE

Summary: Average the set of numbers

Syntax: AVERAGE( { NumberSequence N }+ )

Constraints: At least one Number included. Returns an Error if no Numbers provided.

Semantics: Computes SUM(N) / COUNT(N).

6.18.4 AVERAGEA

Summary: Average values, including values of type Text and Logical.

Syntax: AVERAGEA( { Any N }+ )

Constraints: At least one value included. Returns an Error if no value provided.

Semantics: A variant of the AVERAGE function that includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1 and FALSE is treated as 0. Empty cells are not included. Any N may be of type ReferenceList.

6.18.5 AVERAGEIF

Summary: Average the values of cells in a range that meet a criteria.

Syntax: AVERAGEIF( Reference R ; Criterion C [ ; Reference A ] )

Constraints: Does not accept constant values as reference parameters.

Semantics: If reference A is omitted, averages the values of cells in the reference range R that meet the Criterion C (4.11.8). If reference A is given, averages the values of cells of a range that is constructed using the top left cell of reference A and applying the dimensions, shape and size, of reference R. If no cell in range R matches the Criterion C, an Error is returned. If no Numbers are in the range to be averaged, an Error is returned.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also AVERAGEIFS 6.18.6, COUNTIF 6.13.9, SUMIF 6.16.62, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.18.6 AVERAGEIFS

Summary: Average the values of cells that meet multiple criteria in multiple ranges.

Syntax: AVERAGEIFS( Reference A ; Reference R1 ; Criterion C1 [ ; Reference R2 ; Criterion C2 ]... )

Constraints: Does not accept constant values as reference parameters.

Semantics: Averages the values of cells in the reference range A that meet the Criterion C1 in the reference range R1 and the Criterion C2 in the reference range R2, and so on (4.11.8). All reference ranges shall have the same dimension and size, else an Error is returned. A logical AND is applied between each array result of each selection; a cell of reference range A is evaluated only if the same position in each array is the result of a Criterion match. If no numbers are in the result set to be averaged, an Error is returned.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS or HOST-SEARCH-CRITERIA-MUST-APPLY-TO-WHOLE-CELL properties. 3.4

See also AVERAGEIF 6.18.5, COUNTIFS 6.13.10, SUMIFS 6.16.63, Infix Operator "=" 6.4.7, Infix Operator "<>" 6.4.8, Infix Operator Ordered Comparison ("<", "<=", ">", ">=") 6.4.9

6.18.7 BETADIST

Summary: returns the value of the probability density function or the cumulative distribution function for the beta distribution.

Syntax: BETADIST( Number x ; Number α ; Number β [ ; Number a = 0 [ ; Number b = 1 [ ; Logical Cumulative = TRUE ] ] ] )

Constraints: α > 0, β > 0, a < b,
If α < 1, then the density function has a pole at x = a.
If β < 1, then the density function has a pole at x = b.
In both cases, if x = a respectively x = b and Cumulative = FALSE, an Error is returned.

Semantics: If Cumulative is FALSE, BETADIST returns 0 if x < a or x > b and the value

\frac{Γ (α + β)}{Γ (α) Γ (β)} \cdot {(\frac{x - a}{b - a})}^{α - 1} \cdot {(1 - \frac{x - a}{b - a})}^{β - 1} \cdot \frac{1}{b - a}

If Cumulative is TRUE, BETADIST returns 0 if x < a, 1 if x > b, and the value

\int_{a}^{x} \frac{Γ (α + β)}{Γ (α) Γ (β)} \cdot {(\frac{t - a}{b - a})}^{α - 1} \cdot {(1 - \frac{t - a}{b - a})}^{β - 1} \cdot \frac{1}{b - a} dt

the term can be written as

Note: With substitution

z ≝ \frac{t - a}{b - a}

\int_{0}^{\frac{x - a}{b - a}} \frac{Γ (α + β)}{Γ (α) Γ (β)} \cdot z^{α - 1} \cdot {(1 - z)}^{β - 1} dz

6.18.8 BETAINV

Summary: returns the inverse of BETADIST(x;α;β;A;B;TRUE()).

Syntax: BETAINV( Number P ; Number α ; Number β [ ; Number A = 0 [ ; Number B = 1 ] ] )

Constraints: 0 ≤ P ≤ 1, α > 0, β > 0, A < B

Semantics: BETAINV returns the unique number x in the closed interval from A to B such that BETADIST(x;α;β;A;B) = P.

6.18.9 BINOM.DIST.RANGE

Summary: Returns the probability of a trial result using binomial distribution.

Syntax: BINOM.DIST.RANGE( Integer N ; Number P ; Integer S [ ; Integer S2 ] )

Constraints: 0 ≤ P ≤ 1, 0 ≤ S ≤ S2 ≤ N

Semantics: Let N be a total number of independent trials, and P be a probability of success for each trial. This function returns the probability that the number of successful trials shall be exactly S. If the optional parameter S2 is provided, this function returns the probability that the number of successful trials shall lie between S and S2 inclusive.

This function is computed as follows:

If S2 is not given, let S2 = S. Then the function returns the value of

\sum_{k = S}^{S 2} (\begin{matrix} N \\ k \end{matrix}) P^{k} {(1 - P)}^{N - k}

6.18.10 BINOMDIST

Summary: Returns the binomial distribution.

Syntax: BINOMDIST( Integer S ; Integer N ; Number P ; Logical Cumulative )

Constraints: 0 ≤ P ≤ 1; 0 ≤ S ≤ N

Semantics: If Cumulative is FALSE, this function returns the same result as BINOM.DIST.RANGE(N;P;S). If Cumulative is TRUE, it is equivalent to calling BINOM.DIST.RANGE(N;P;0;S).

6.18.11 LEGACY.CHIDIST

Summary: returns the right-tail probability for the χ2-distribution.

Syntax: LEGACY.CHIDIST( Number X ; Number DegreesOfFreedom )

Constraints: DegreesOfFreedom is a positive integer.

Semantics: In the following n is DegreesOfFreedom. LEGACY.CHIDIST returns 1 for X ≤ 0 and the value

\int_{X}^{\infty} \frac{t^{\frac{n}{2} - 1} e^{- \frac{t}{2}}}{2^{\frac{n}{2}} Γ (n / 2)} dt

for X > 0.

See also CHISQDIST 6.18.12, LEGACY.CHITEST 6.18.15

6.18.12 CHISQDIST

Summary: returns the value of the probability density function or the cumulative distribution function for the χ2-distribution.

Syntax: CHISQDIST( Number X ; Number DegreesOfFreedom [ ; Logical Cumulative = TRUE ] )

Constraints: DegreesOfFreedom is a positive integer.

Semantics: In the following n is DegreesOfFreedom.

If Cumulative is FALSE, CHISQDIST returns 0 for X ≤ 0 and the value

\frac{X^{\frac{n}{2} - 1} e^{- \frac{X}{2}}}{2^{\frac{n}{2}} Γ (n / 2)}

for X > 0.

If Cumulative is TRUE, CHISQDIST returns 0 for X ≤ 0 and the value

\int_{0}^{X} \frac{t^{\frac{n}{2} - 1} e^{- \frac{t}{2}}}{2^{\frac{n}{2}} Γ (n / 2)} dt

for X > 0.

6.18.13 LEGACY.CHIINV

Summary: returns the inverse of LEGACY.CHIDIST(x; DegreesOfFreedom).

Syntax: LEGACY.CHIINV( Number P ; Number DegreesOfFreedom )

Constraints: DegreesOfFreedom is a positive integer and 0 < P ≤ 1.

Semantics: LEGACY.CHIINV returns the unique number x such that LEGACY.CHIDIST(x; DegreesOfFreedom) = P.

6.18.14 CHISQINV

Summary: returns the inverse of CHISQDIST(x; DegreesOfFreedom; TRUE()).

Syntax: CHISQINV( Number P ; Number DegreesOfFreedom )

Constraints: DegreesOfFreedom is a positive integer and 0 < P ≤ 1 .

Semantics: CHISQINV returns the unique number x ≥ 0 such that CHISQDIST(x; DegreesOfFreedom;TRUE()) = P.

6.18.15 LEGACY.CHITEST

Summary: Returns some Chi square goodness-for-fit test.

Syntax: LEGACY.CHITEST( ForceArray Array A ; ForceArray Array E )

Constraints:
ROWS(A) = ROWS(E)
COLUMNS(A) = COLUMNS(E)
COLUMNS(A) * ROWS(A) > 1

For an empty element or an element of type Text or Boolean in A the element at the corresponding position of E is ignored, and vice versa.

•A: actual observation data.
•E: expected values.

First a Chi square statistic is calculated:

\begin{matrix} χ^{2} = \sum_{i = 1}^{r} \sum_{j = 1}^{c} \frac{{(A_{ij} - E_{ij})}^{2}}{E_{ij}} \\ with \\ r = number of rows \\ c = number of columns \\ A_{ij} = element of actual data \\ E_{ij} = element of expected values \end{matrix}

Then LEGACY.CHIDIST is called with the Chi-square value and a degree of freedom (df):

\begin{matrix} if r > 1 and c > 1 \\ df = (r - 1) \cdot (c - 1) \\ else \\ df = r \cdot c - 1 \\ LEGACY . CHITEST = LEGACY . CHIDIST (χ^{2}; df) \end{matrix}

See also COLUMNS 6.13.5, ROWS 6.13.30, LEGACY.CHIDIST 6.18.11

6.18.16 CONFIDENCE

Summary: Returns the confidence interval for a population mean.

Syntax: CONFIDENCE( Number Alpha ; Number Stddev ; Number Size )

Constraints: 0 < Alpha < 1; Stddev > 0, Size ≥ 1

Semantics: Calling this function is equivalent to calling
NORMINV(1 - Alpha / 2; 0; 1) * Stddev / SQRT (Size)

6.18.17 CORREL

Summary: Calculates the correlation coefficient of values in N1 and N2.

Syntax: CORREL( ForceArray Array N1 ; ForceArray Array N2 )

Constraints: COLUMNS(N1) = COLUMNS(N2), ROWS(N1) = ROWS(N2), both sequences shall contain at least one number at corresponding positions each.

Semantics: Has the same value as COVAR(N1;N2) / STDEVP(N1) * (STDEVP(N2)). The CORREL function actually is identical to the PEARSON function.

For an empty element or an element of type Text or Boolean in N1 the element at the corresponding position of N2 is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30, COVAR 6.18.18, STDEVP 6.18.74, PEARSON 6.18.56

6.18.18 COVAR

Summary: Calculates covariance of two cell ranges.

Syntax: COVAR( ForceArray Array N1 ; ForceArray Array N2 )

Constraints: COLUMNS(N1) = COLUMNS(N2), ROWS(N1) = ROWS(N2), both sequences shall contain at least one number at corresponding positions each.

Semantics: returns

\frac{1}{N} \cdot (\sum_{a \in N 1, b \in N 2} (a - \bar{N 1}) \cdot (b - \bar{N 2}))

is the result of calling AVERAGE(

\bar{N 1}

), and

\bar{N 2}

is the result of calling AVERAGE(

), and N is the number of terms in the sum.

For an empty element or an element of type Text or Boolean in N1 the element at the corresponding position of N2 is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30, AVERAGE 6.18.3

6.18.19 CRITBINOM

Summary: Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value.

Syntax: CRITBINOM( Number Trials ; Number SP ; Number Alpha )

Constraints: Trials ≥ 0, 0 ≤ SP ≤ 1, 0 ≤ Alpha ≤ 1

Summary: Calculates sum of squares of deviations.

•Trials: the total number of trials.
•SP: the probability of success for one trial.
•Alpha: the threshold probability to be reached or exceeded.

6.18.20 DEVSQ

Syntax: DEVSQ( { NumberSequence N }+ )

where a is the result of calling AVERAGE(N).

Semantics: returns

\sum_{x \in N} {(x - a)}^{2}

6.18.21 EXPONDIST

Summary: returns the value of the probability density function or the cumulative distribution function for the exponential distribution.

Syntax: EXPONDIST( Number X ; Number λ [ ; Logical Cumulative = TRUE ] )

Semantics: If Cumulative is FALSE, EXPONDIST returns 0 if X < 0 and the value

Constraints: λ > 0

λ e^{- λ X}

If Cumulative is TRUE, EXPONDIST returns 0 if X < 0 and the value

\int_{0}^{X} λ e^{- λ t} dt = 1 - e^{- λ X}

Summary: returns the value of the probability density function or the cumulative distribution function for the F-distribution.

6.18.22 FDIST

Syntax: FDIST( Number X ; Number R1 ; Number R2 [ ; Logical Cumulative = TRUE ] )

Constraints: R1 and R2 are positive integers

If Cumulative is FALSE, FDIST returns 0 if X < 0, an Error if the numerator degrees of freedom R1 = 1 and X = 0, and the value

•R1: the degrees of freedom in the numerator of the F distribution.
•R2: the degrees of freedom in the denominator of the F distribution.

\frac{Γ (\frac{R_{1} + R_{2}}{2}) {(\frac{R_{1}}{R_{2}})}^{\frac{R_{1}}{2}}}{Γ (\frac{R_{1}}{2}) Γ (\frac{R_{2}}{2})} \cdot \frac{x^{\frac{R_{1}}{2} - 1}}{{(1 + \frac{R_{1}}{R_{2}} x)}^{\frac{R_{1} + R_{2}}{2}}}

If the numerator degrees of freedom

= 1, then the density function has a pole at

= 0, the subterm

x^{\frac{R_{1}}{2} - 1} = 0^{- 0.5}

is not defined.

If Cumulative is TRUE, FDIST returns 0 if X < 0 and the value

\frac{Γ (\frac{R_{1} + R_{2}}{2}) {(\frac{R_{1}}{R_{2}})}^{\frac{R_{1}}{2}}}{Γ (\frac{R_{1}}{2}) Γ (\frac{R_{2}}{2})} \cdot \int_{0}^{X} \frac{t^{\frac{R_{1}}{2} - 1}}{{(1 + \frac{R_{1}}{R_{2}} t)}^{\frac{R_{1} + R_{2}}{2}}} dt

Summary: returns the area of the right tail of the probability density function for the F-distribution.

6.18.23 LEGACY.FDIST

Syntax: LEGACY.FDIST( Number X ; Number R1 ; Number R2 )

Constraints: R1 and R2 are positive integers

LEGACY.FDIST returns Error if x < 0 and the value

\frac{Γ (\frac{r_{1} + r_{2}}{2}) {(\frac{r_{1}}{r_{2}})}^{\frac{r_{1}}{2}}}{Γ (\frac{r_{1}}{2}) Γ (\frac{r_{2}}{2})} \cdot \int_{x}^{\infty} \frac{t^{\frac{r_{1}}{2} - 1}}{{(1 + \frac{r_{1}}{r_{2}} t)}^{\frac{r_{1} + r_{2}}{2}}} dt

Note that the latter is (1-FDIST(x; r1; r2;TRUE())).

6.18.24 FINV

Summary: returns the inverse of FDIST(x;R1;R2;TRUE()).

Syntax: FINV( Number P ; Number R1 ; Number R2 )

Constraints: 0 ≤ P < 1, R1 and R2 are positive integers

Semantics: FINV returns the unique non-negative number x such that FDIST(x;R1;R2) = P.

See also FDIST 6.18.22, LEGACY.FDIST 6.18.23, LEGACY.FINV 6.18.25

6.18.25 LEGACY.FINV

Summary: returns the inverse of LEGACY.FDIST(x;R1;R2).

Syntax: LEGACY.FINV( Number P ; Number R1 ; Number R2 )

Constraints: 0 < P ≤ 1, R1 and R2 are positive integers

Semantics: LEGACY.FINV returns the unique non-negative number x such that LEGACY.FDIST(x;R1;R2) = P.

See also FDIST 6.18.22, LEGACY.FDIST 6.18.23, FINV 6.18.24

6.18.26 FISHER

Summary: returns the Fisher transformation.

Syntax: FISHER( Number R )

Semantics: Returns the Fisher transformation with a sample correlation R. This function computes

Constraints: -1 < R < 1

\frac{1}{2} \ln (\frac{1 + R}{1 - R})

where ln is the natural logarithm function.

FISHER is a synonym for ATANH.

6.18.27 FISHERINV

Summary: returns the inverse Fisher transformation.

Syntax: FISHERINV( Number R )

Semantics: Returns the inverse Fisher transformation. This function computes

Constraints: none

\frac{e^{2 R} - 1}{e^{2 R} + 1}

FISHERINV is a synonym for TANH.

6.18.28 FORECAST

Summary: Extrapolates future values based on existing x and y values.

Syntax: FORECAST( Number Value ; ForceArray Array Data_Y ; ForceArray Array Data_X )

Constraints: COLUMNS(Data_Y) = COLUMNS(Data_X), ROWS(Data_Y) = ROWS(Data_X)

For an empty element or an element of type Text or Boolean in Data_Y the element at the corresponding position of Data_X is ignored, and vice versa.

•Value: the x-value, for which the y-value on the linear regression is to be returned.
•Data_Y: the array or range of known y-values.
•Data_X: the array or range of known x-values.

See also COLUMNS 6.13.5, ROWS 6.13.30

6.18.29 FREQUENCY

Summary: Categorizes values into intervals and counts the number of values in each interval.

Syntax: FREQUENCY( NumberSequenceList Data ; NumberSequenceList Bins )

Constraints: Values in Bins shall be sorted in ascending order and Bins shall be a column vector. Evaluators may accept unsorted values in bins.

Semantics: Counts the number of values for each interval given by the border values in Bins .
The values in Bins determine the upper boundaries of the intervals. The intervals include the upper boundarie. The returned array is a column vector and has one more element than Bins ; the last element represents the number of all elements greater than the last value in Bins . If Bins is empty, all values in Data are counted. The values in the result array are ordered matching the original order of Bins . If the values in Bins are not sorted in ascending order, they are sorted internally to form category intervals and the counts of Data values are "unsorted" to the original order of Bins. If Data is empty, the value of all elements in the returned array is 0.

Data: The data, that should be categorized and counted according to the given intervals.
Bins: The upper boundaries determining the intervals the values in data should be grouped by.

6.18.30 FTEST

Summary: Calculates the probability of an F-test.

Syntax: FTEST( ForceArray NumberSequence Data_1 ; ForceArray NumberSequence Data_2 )

Constraints: Data_1 and Data_2 shall both contain at least 2 numbers and shall both have nonzero variances

Calculates a two-sided P-value to decide, whether the difference in the variances of the two data sets are significant enough to reject the hypothesise, that both sets come from normally distributed populations with the same variances.

Suppose the data set Data_1 is a sample of size

n_{1}

from a normal distribution and has the sample variance

{s_{1}}^{2}

, and the data set Data_2 is a sample of size

n_{2}

from a normal distribution and has the sample variance

{s_{2}}^{2}

Get the value

P_{tail}

as the area of the right tail beyond

{s_{1}}^{2} / {s_{2}}^{2}

of the F‑distribution with numerator degrees of freedom

n_{1} - 1

and denominator degrees of freedom

n_{2} - 1

FTEST returns twice the minimum of the values

P_{tail}

Summary: returns the value of the probability density function or the cumulative distribution function for the Gamma distribution.

1 - P_{tail}

.See also TTEST 6.18.81

6.18.31 GAMMADIST

Syntax: GAMMADIST( Number X ; Number α ; Number β [ ; Logical Cumulative = TRUE ] )

Constraints: α > 0, β > 0

Semantics: If Cumulative is FALSE, GAMMADIST returns 0 if X < 0 and the value

\frac{1}{β^{α} \cdot Γ (α)} \cdot X^{α - 1} \cdot e^{- \frac{X}{β}}

If Cumulative is TRUE(), GAMMADIST returns 0 if X < 0 and the value

\int_{0}^{X} \frac{1}{β^{α} \cdot Γ (α)} \cdot t^{α - 1} \cdot e^{- \frac{t}{β}} dt

Summary: returns the inverse of GAMMADIST(X;α;β;TRUE).

6.18.32 GAMMAINV

Syntax: GAMMAINV( Number P ; Number α ; Number β )

Constraints: 0 ≤ P < 1, α > 0, β > 0

Semantics: GAMMAINV returns the unique number X ≥ 0 such that GAMMAINV(X;α;β) = P.

6.18.33 GAUSS

Summary: Returns 0.5 less than the standard normal cumulative distribution

Syntax: GAUSS( Number X )

Semantics: Returns NORMDIST(X;0;1;TRUE())-0.5

6.18.34 GEOMEAN

Summary: returns the geometric mean of a sequence

Syntax: GEOMEAN( { NumberSequenceList N }+ )

Semantics: Returns the geometric mean of a given sequence. That means

{(\prod_{a \in N} a)}^{1 / n}

where n is a result of calling COUNT(N).

6.18.35 GROWTH

Summary: Returns predicted values based on an exponential regression.

Syntax: GROWTH( Array KnownY [ ; [ Array KnownX ] [ ; [ Array NewX ] [ ; Logical Const = TRUE ] ] ] )

Constraints: (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX)) or (COLUMNS(KnownY) = 1 and ROWS(KnownY) = ROWS(KnownX) and COLUMNS(KnownX) = COLUMNS(NewX)) or (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = 1 and ROWS(KnownX) = ROWS(NewX))

k = ROWS(KnownY) ∙ COLUMNS(KnownY)

•KnownY: The set of known y-values to be used to determine the regression equation
•KnownX: The set of known x-values to be used to determine the regression equation. If omitted or an empty parameter, it is set to the sequence 1,2,3,…,k , where

•NewX: The set of x-values for which predicted y-values are to be calculated. If omitted or an empty parameter, it is set to KnownX.

Const: If set to FALSE, the model constant a is equal to 0.

LOGEST(KnownY ; KnownX; Const; FALSE) either returns an error or an array with 1 row and n+1 columns. If it returns an error then so does GROWTH. If it returns an array, we call the entries in that array

b_{n}, b_{n - 1}, \dots, b_{1}, a

Let

z_{i j}

denote the entry in the ith row and jth column of NewX.

If COLUMNS(KnownY ) = COLUMNS(KnownX) and ROWS(KnownY ) = ROWS(KnownX), then GROWTH returns an array with ROWS(NewX) rows and COLUMNS(NewX) column, such that the entry in its ith row and jth column is

a \times b_{1}^{z_{i j}}

Otherwise, if COLUMNS(KnownY ) = 1 and ROWS(KnownY ) = ROWS(KnownX) and COLUMNS(KnownX) = COLUMNS(NewX), then GROWTH returns an array with ROWS(NewX) rows and 1 column, such that the entry in the ith row is

a \times \prod_{j = 1}^{n} {b_{j}}^{z_{i j}}

Otherwise, if COLUMNS(KnownY ) = COLUMNS(KnownX) and ROWS(KnownY ) = 1 and ROWS(KnownX) = ROWS(NewX), then GROWTH returns an array with 1 row and COLUMNS(NewX) columns, such that the entry in the jth column is

a \times \prod_{i = 1}^{n} {b_{i}}^{z_{i j}}

See also COLUMNS 6.13.5, ROWS 6.13.30, LOGEST 6.18.42, TREND 6.18.79

6.18.36 HARMEAN

Summary: returns the harmonic mean of a sequence

Syntax: HARMEAN( { NumberSequenceList N }+ )

Semantics: Returns the harmonic mean of a given sequence. That means

\frac{n}{\sum_{i = 1}^{n} \frac{1}{a_{i}}}

where a1,a2,...,an are the numbers of the sequence N and n is a result of calling COUNT(N).

6.18.37 HYPGEOMDIST

Summary: The hypergeometric distribution returns the number of successes in a sequence of n draws from a finite population without replacement.

Syntax: HYPGEOMDIST( Integer X ; Integer T ; Integer M ; Integer N [ ; Logical Cumulative = FALSE ] )

Constraints: 0 ≤ X ≤ T ≤ N, 0 ≤ M ≤ N

If Cumulative is FALSE, return the probability of exactly X successes. If Cumulative is TRUE, return the probability of at most X successes. If omitted, FALSE is assumed.

•X: the number of successes in T trials
•T: the number of trials
•M: the number of successes in the population
•N: the total population
•Cumulative : a Logical parameter.

If Cumulative is FALSE, HYPGEOMDIST returns

\frac{(\begin{matrix} M \\ X \end{matrix}) (\begin{matrix} N - M \\ T - X \end{matrix})}{(\begin{matrix} N \\ T \end{matrix})}

If Cumulative is TRUE, HYPGEOMDIST returns

\sum_{i = 0}^{X} \frac{(\begin{matrix} M \\ i \end{matrix}) (\begin{matrix} N - M \\ T - i \end{matrix})}{(\begin{matrix} N \\ T \end{matrix})}

Note:

(\begin{matrix} x \\ y \end{matrix}) = 0 for y > x

6.18.38 INTERCEPT

Summary: Returns the y-intercept of the linear regression line for the given data.

Syntax: INTERCEPT( ForceArray Array Data_Y ; ForceArray Array Data_X )

Constraints: COLUMNS(Data_X) = COLUMNS(Data_Y), ROWS(Data_X) = ROWS(Data_Y)

Summary: Return the kurtosis (“peakedness”) of a data set.

INTERCEPT returns the intercept (a) calculated as described in 6.18.41 for the function call LINEST(Data_Y,Data_X,FALSE()).

For an empty element or an element of type Text or Boolean in Data_Y the element at the corresponding position of Data_X is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30

6.18.39 KURT

Syntax: KURT( { NumberSequenceList X }+ )

Constraints: COUNT(X) ≥ 4, STDEV(X) ≠ 0

Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. Positive kurtosis indicates a relatively peaked distribution (compared to the normal distribution), while negative kurtosis indicates a relatively flat distribution.

kurtosis = (\frac{n (n + 1)}{(n - 1) (n - 2) (n - 3)} \sum_{i = 1}^{n} {(\frac{x_{i} - \bar{x}}{s})}^{4}) - \frac{3 {(n - 1)}^{2}}{(n - 2) (n - 3)}

where s is the sample standard deviation, and n is the number of numbers.

6.18.40 LARGE

Summary: Finds the nth largest value in a list.

Syntax: LARGE( NumberSequenceList List ; Number|Array N )

Returns: Number or Array

Constraints: ROUNDUP(N;0) = N. If the resulting N is <1 or larger than the size of List, Error is returned

Semantics: If N is an array of numbers, an array of largest values is returned.

6.18.41 LINEST

Summary: Returns the parameters of the (simple or multiple) linear regression equation for the given data and, optionally, statistics on this regression.

Syntax: LINEST( ForceArray Array KnownY [ ; [ ForceArray Array KnownX ] [ ; Logical Const = TRUE [ ; Logical Stats = FALSE ] ] ] )

Constraints: (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX)) or (COLUMNS(KnownY) = 1 and ROWS(KnownY) = ROWS(KnownX)) or (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = 1)

Const: If set to FALSE, the model constant a is equal to 0.

•KnownY: The set of y-values for the equation
•KnownX: The set of x-values for the equation. If omitted or an empty parameter, it is set to the sequence 1,2,3,…,k , where k = ROWS(KnownY) ∙ COLUMNS(KnownY).

•Stats: If FALSE, only the regression coefficient is to be calculated. If set to TRUE, the result will include other statistical data.

If any of the entries in KnownY and KnownX do not convert to Number, LINEST returns an error.

The result created by LINEST if STATS is TRUE is given in Table 29 - LINEST. If STATS is FALSE it is just the first row of Table 29 - LINEST. The empty cells in this table are returned as empty or as containing an error.

Table 29 - LINEST

bn	bn-1	…	b1	a
$s_{b_{n}}$	$s_{b_{n - 1}}$	…	$s_{b_{1}}$	$s_{a}$
$R^{2}$	$s_{e}$
F	df
SSreg	SSresid

If COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX) then n = 1, k = ROWS(KnownY) ∙ COLUMNS(KnownY), the entries of KnownX in column major order are denoted with

x_{1 n}, x_{2 n}, \dots, x_{k n}

and the entries of KnownY in column major order are denoted with

y_{1}, y_{2}, \dots, y_{k}

Otherwise but if COLUMNS(KnownY) = 1, then n = COLUMNS(KnownX), k = ROWS(KnownY), the entry in the jth column and ith row of KnownX is denoted

x_{i j}

and the entry in the ith row of KnownY is denoted

y_{i}

Otherwise but if ROWS(KnownY) = 1, then n = ROWS(KnownX), k = COLUMNS(KnownY), the entry in the jth column and ith row of KnownX is denoted

x_{j i}

and the entry in the jth column of KnownY is denoted

y_{j}

If Const is TRUE and k ≤ n + 1, LINEST returns an error. Similarly, if Const is FALSE and k ≤ n, LINEST returns an error.

We denote

\bar{x_{i}} = \frac{1}{k} \cdot \sum_{j = 1}^{k} x_{i j}

and define the following matrices:

\bar{y} = \frac{1}{k} \cdot \sum_{j = 1}^{k} y_{j}

Y = (\begin{matrix} y_{1} \\ ⋮ \\ y_{k} \end{matrix})

Summary: Returns the parameters of an exponential regression equation for the given data obtained by linearizing this intrinsically linear response function and returns, optionally, statistics on this regression.

X = (\begin{matrix} 1 & x_{11} & \dots & x_{1 n} \\ 1 & x_{12} & \dots & x_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & x_{k 1} & \dots & x_{k n} \end{matrix})

for

Const

being TRUE, and

X = (\begin{matrix} x_{11} & \dots & x_{1 n} \\ x_{12} & \dots & x_{2 n} \\ ⋮ & ⋱ & ⋮ \\ x_{k 1} & \dots & x_{k n} \end{matrix})

X^{T}

denote the transpose of

, see TRANSPOSE

6.5.6

. Then the matrix product

X^{T} \cdot X

is a square matrix. If

X^{T} \cdot X

is not invertible, then LINEST shall either return an error or calculate a result as described below.

X^{T} \cdot X

is invertible, then

{(X^{T} \cdot X)}^{- 1} \cdot X^{T} \cdot Y

is a matrix

with a single column. If

Const

is TRUE, the entries of

are denoted

B = (\begin{matrix} a \\ b_{1} \\ ⋮ \\ b_{n} \end{matrix})

; if

Const

is FALSE, the entries of

are denoted

B = (\begin{matrix} b_{1} \\ ⋮ \\ b_{n} \end{matrix})

and

= 0.

These

a, b_{1}, \dots, b_{n}

are the values returned by LINEST in the first row of its result array in the order given in

Table 29 - LINEST

The statistics in the 2nd to 5th rows of Table 29 - LINEST are as follows:

If Const is TRUE:

df = k - n - 1

{SS}_{resid} = \sum_{i = 1}^{k} {((a + \sum_{j = 1}^{n} b_{j} x_{i j}) - y_{i})}^{2}

{SS}_{reg} = \sum_{i = 1}^{k} {((a + \sum_{j = 1}^{n} b_{j} x_{i j}) - \bar{y})}^{2}

s_{a} = s_{e} \sqrt{d_{1}}

and

s_{b_{i}} = s_{e} \sqrt{d_{i + 1}}

where

d_{i}

is the element in the

th row and

th column of

{(X^{T} \cdot X)}^{- 1}

s_{e} = \sqrt{\frac{{SS}_{resid}}{df}}

R^{2} = \frac{{SS}_{reg}}{\sum_{i = 1}^{k} {(y_{i} - \bar{y})}^{2}}

and

F = \frac{{SS}_{reg} / n}{{SS}_{resid} / df}

If Const is FALSE:

df = k - n

{SS}_{resid} = \sum_{i = 1}^{k} {((\sum_{j = 1}^{n} b_{j} x_{i j}) - y_{i})}^{2}

{SS}_{reg} = \sum_{i = 1}^{k} {(\sum_{j = 1}^{n} b_{j} x_{i j})}^{2}

s_{b_{i}} = s_{e} \sqrt{d_{i}}

where

d_{i}

is the element in the

th row and

th column of

{(X^{T} \cdot X)}^{- 1}

s_{e} = \sqrt{\frac{{SS}_{resid}}{df}}

R^{2} = \frac{{SS}_{reg}}{\sum_{i = 1}^{k} y_{i}^{2}}

and

F = \frac{{SS}_{reg} / n}{{SS}_{resid} / df}

In this case

s_{a}

is undefined and is returned as either 0, blank or an error.

X^{T} \cdot X

is not invertible, then the columns of X are linearly dependent. In this case an evaluator shall return an error or select any maximal linearly independent subset of these columns that if

Const

is TRUE includes the first column and perform the above calculations with that subset. In the latter case the coefficients

b_{i}

of omitted columns are returned as 0.

See also COLUMNS 6.13.5, ROWS 6.13.30

6.18.42 LOGEST

Syntax: LOGEST( ForceArray Array KnownY [ ; [ ForceArray Array KnownX ] [ ; Logical Const = TRUE [ ; Logical Stats = FALSE ] ] ] )

Constraints: (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX)) or (COLUMNS(KnownY) = 1 and ROWS(KnownY) = ROWS(KnownX)) or (COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = 1)

Const: If set to FALSE, the model constant a is equal to 0.

•KnownY: The set of y-values for the equation
•KnownX: The set of x-values for the equation. If omitted or an empty parameter, it is set to the sequence 1,2,3,…,k, where k = ROWS(KnownY) ∙ COLUMNS(KnownY).

•Stats: If FALSE, only the regression coefficient is to be calculated. If set to TRUE, the result will include other statistical data.

If any of the entries in KnownY and KnownX do not convert to Number or if any of the entries in KnownY is negative, LOGEST returns an error.

The result created by LOGEST if STATS is TRUE is given in Table 30 - LOGEST. If STATS is FALSE it is just the first row of Table 30 - LOGEST. The empty cells in this table are returned as empty or as containing an error.

Table 30 - LOGEST

$e^{b_{n}}$	$e^{b_{n - 1}}$	…	$e^{b_{1}}$	$e^{a}$
$s_{b_{n}}$	$s_{b_{n - 1}}$	…	$s_{b_{1}}$	$s_{a}$
$R^{2}$	$s_{e}$
F	df
SSreg	SSresid

If COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX) then n = 1, k = ROWS(KnownY) ∙ COLUMNS(KnownY), the entries of KnownX in column major order are denoted with

x_{1 n}, x_{2 n}, \dots, x_{k n}

and the entries of KnownY in column major order are denoted with

y_{1}, y_{2}, \dots, y_{k}

Otherwise but if COLUMNS(KnownY) = 1, then n = COLUMNS(KnownX), k = ROWS(KnownY), the entry in the jth column and ith row of KnownX is denoted

x_{i j}

and the entry in the ith row of KnownY is denoted

y_{i}

Otherwise but if ROWS(KnownY) = 1, then n = ROWS(KnownX), k = COLUMNS(KnownY), the entry in the jth column and ith row of KnownX is denoted

x_{j i}

and the entry in the jth column of KnownY is denoted

y_{j}

If Const is TRUE and k ≤ n + 1, LOGEST returns an error. Similarly, if Const is FALSE and k ≤ n, LOGEST returns an error.

We denote

\bar{x_{i}} = \frac{1}{k} \cdot \sum_{j = 1}^{k} x_{i j}

and define the following matrices:

\bar{\ln (y)} = \frac{1}{k} \cdot \sum_{j = 1}^{k} \ln (y_{j})

Y = (\begin{matrix} \ln (y_{1}) \\ ⋮ \\ \ln (y_{k}) \end{matrix})

Summary: returns the inverse of LOGNORMDIST(x;Mean;StandardDeviation,TRUE()).

X = (\begin{matrix} 1 & x_{11} & \dots & x_{1 n} \\ 1 & x_{12} & \dots & x_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & x_{k 1} & \dots & x_{k n} \end{matrix})

for

Const

being TRUE, and

X = (\begin{matrix} x_{11} & \dots & x_{1 n} \\ x_{12} & \dots & x_{2 n} \\ ⋮ & ⋱ & ⋮ \\ x_{k 1} & \dots & x_{k n} \end{matrix})

X^{T}

denote the transpose of

, see TRANSPOSE

6.5.6

. Then the matrix product

X^{T} \cdot X

is a square matrix. If

X^{T} \cdot X

is not invertible, then LOGEST shall either return an error or calculate a result as described below.

X^{T} \cdot X

is invertible, then

{(X^{T} \cdot X)}^{- 1} \cdot X^{T} \cdot Y

is a matrix

with a single column. If

Const

is TRUE, the entries of

are denoted

B = (\begin{matrix} a \\ b_{1} \\ ⋮ \\ b_{n} \end{matrix})

; if

Const

is FALSE, the entries of

are denoted

B = (\begin{matrix} b_{1} \\ ⋮ \\ b_{n} \end{matrix})

and

= 0.

Then

e^{a}, e^{b_{1}}, \dots, e^{b_{n}}

are the values returned by LOGEST in the first row of its result array in the order given in

Table 1 - Operators

The statistics in the 2nd to 5th rows of Table 1 - Operators are as follows:

If Const is TRUE():

df = k - n - 1

{SS}_{resid} = \sum_{j = 1}^{k} {((a + \sum_{i = 1}^{n} b_{j} x_{i j}) - \ln (y_{i}))}^{2}

{SS}_{reg} = \sum_{j = 1}^{k} {((a + \sum_{i = 1}^{n} b_{j} x_{i j}) - \bar{\ln (y)})}^{2}

s_{a} = s_{e} \sqrt{d_{1}}

and

s_{b_{i}} = s_{e} \sqrt{d_{i + 1}}

where

d_{i}

is the element in the

th row and

th column of

{(X^{T} \cdot X)}^{- 1}

s_{e} = \sqrt{\frac{{SS}_{resid}}{df}}

R^{2} = \frac{{SS}_{reg}}{\sum_{j = 1}^{k} {(\ln (y_{k}) - \bar{\ln (y)})}^{2}}

and

F = \frac{{SS}_{reg} / n}{{SS}_{resid} / df}

If Const is FALSE:

df = k - n

{SS}_{resid} = \sum_{j = 1}^{k} {((\sum_{i = 1}^{n} b_{j} x_{i j}) - \ln (y_{i}))}^{2}

{SS}_{reg} = \sum_{j = 1}^{k} {(\sum_{i = 1}^{n} b_{j} x_{i j})}^{2}

s_{b_{i}} = s_{e} \sqrt{d_{i}}

where

d_{i}

is the element in the

th row and

th column of

{(X^{T} \cdot X)}^{- 1}

s_{e} = \sqrt{\frac{{SS}_{resid}}{df}}

R^{2} = \frac{{SS}_{reg}}{\sum_{j = 1}^{k} \ln {(y_{k})}^{2}}

and

F = \frac{{SS}_{reg} / n}{{SS}_{resid} / df}

In this case

s_{a}

is undefined and is returned as either 0, blank or an error.

X^{T} \cdot X

is not invertible, then the columns of X are linearly dependent. In this case an evaluator shall return an error or select any maximal linearly independent subset of these columns that if Const is TRUE includes the first column and perform the above calculations with that subset. In the latter case the coefficients

e^{b_{i}}

of omitted columns are returned as 1.

See also COLUMNS 6.13.5, ROWS 6.13.30

6.18.43 LOGINV

Syntax: LOGINV( Number P [ ; Number Mean = 0 [ ; Number StandardDeviation = 1 ] ] )

Constraints: StandardDeviation > 0 and 0 < P < 1.

Semantics: LOGINV returns the unique number x such that LOGNORMDIST(x;Mean;StandardDeviation;TRUE()) = P.

6.18.44 LOGNORMDIST

Summary: returns the value of the probability density function or the cumulative distribution function for the lognormal distribution with the mean and standard deviation given.

Syntax: LOGNORMDIST( Number X [ ; Number μ = 0 [ ; Number σ = 1 [ ; Logical Cumulative = TRUE ] ] ] )

Constraints: σ > 0; X > 0 if Cumulative is FALSE

Semantics: If Cumulative is FALSE, LOGNORMDIST returns the value

\frac{e^{- \frac{1}{2} \cdot {(\frac{\ln (X) - μ}{σ})}^{2}}}{X \sqrt{2 π} σ}

If Cumulative is TRUE, LOGNORMDIST returns the value

\int_{0}^{X} \frac{e^{- \frac{1}{2} \cdot {(\frac{\ln (t) - μ}{σ})}^{2}}}{t \sqrt{2 π} σ} dt

if X > 0 and 0 otherwise.

6.18.45 MAX

Summary: Return the maximum from a set of numbers.

Syntax: MAX( { NumberSequenceList N }+ )

Semantics: Returns the value of the maximum number in the list passed in. Non-numbers are ignored. Note that if Logical types are a distinct type, they are not included.

6.18.46 MAXA

Summary: Return the maximum from a set of values, including values of type Text and Logical.

Syntax: MAXA( { Any N }+ )

Semantics: A variation of the MAX function that includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1, and FALSE is treated as 0. Empty cells are not included. Any N may be of type ReferenceList.

6.18.47 MEDIAN

Summary: Returns the median (middle) value in the list.

Syntax: MEDIAN( { NumberSequenceList X }+ )

MEDIAN logically ranks the numbers (lowest to highest). If given an odd number of values, MEDIAN returns the middle value. If given an even number of values, MEDIAN returns the arithmetic average of the two middle values.

\begin{matrix} n = is the count of the ranked numbersequence \\ \tilde{x} = x_{(\frac{(n + 1)}{2})} \\ for n = odd \\ \tilde{x} = \frac{1}{2} (x_{(\frac{n}{2})} + x_{(\frac{n}{2} + 1)}) \\ for n = even \end{matrix}

6.18.48 MIN

Summary: Return the minimum from a set of numbers.

Syntax: MIN( { NumberSequenceList N }+ )

Semantics: Returns the value of the minimum number in the list passed in. Returns zero if no numbers are provided in the list. What happens when MIN is provided 0 parameters is implementation-defined, but MIN() with no parameters should return 0.

6.18.49 MINA

Summary: Return the minimum from a set of values, including values of type Text and Logical.

Syntax: MINA( { Any N }+ )

Semantics: A variation of the MIN function that includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1, and FALSE is treated as 0. Empty cells are not included. What happens when MINA is provided 0 parameters is implementation-defined. Any N may be of type ReferenceList.

6.18.50 MODE

Summary: Returns the most common value in a data set.

Syntax: MODE( { ForceArray NumberSequence N }+ )

Semantics: Returns the most common value in a data set. If there are more than one values with the same largest frequency, returns the smallest value. If the number sequence does no contain at least two equal values, the MODE is not defined, as no most common value can be found, and an Error is returned.

6.18.51 NEGBINOMDIST

Summary: Returns the negative binomial distribution.

Syntax: NEGBINOMDIST( Integer X ; Integer R ; Number Prob )

•X: The number of failures.
•R: The threshold number of successes.
•Prob: The probability of a success.

•If (X + R - 1) ≤ 0, NEGBINOMDIST returns an Error.
•If Prob < 0 or Prob > 1, NEGBINOMDIST returns an Error.

NEGBINOMDIST returns the probability that there will be X failures before the R-th success, when the constant probability of a success is Prob.

Note: This function is similar to the binomial distribution, except that the number of successes is fixed, and the number of trials is variable. Like the binomial, trials are assumed to be independent.

\begin{matrix} P_{R, Prob} (X) = (\begin{matrix} X + R - 1 \\ R - 1 \end{matrix}) {Prob}^{R} {(1 - Prob)}^{X} \\ (\begin{matrix} X + R - 1 \\ R - 1 \end{matrix}) is a binomial coefficient \end{matrix}

6.18.52 NORMDIST

Summary: returns the value of the probability density function or the cumulative distribution function for the normal distribution with the mean and standard deviation given.

Syntax: NORMDIST( Number X ; Number Mean ; Number StandardDeviation [ ; Logical Cumulative = TRUE() ] )

Constraints: StandardDeviation > 0.

Semantics: In the following μ is Mean and σ is StandardDeviation.

If Cumulative is FALSE, NORMDIST returns the value

\frac{e^{- \frac{1}{2} \cdot {(\frac{X - μ}{σ})}^{2}}}{\sqrt{2 π} σ}

If Cumulative is TRUE, NORMDIST returns the value

\int_{- \infty}^{X} \frac{e^{- \frac{1}{2} \cdot {(\frac{t - μ}{σ})}^{2}}}{\sqrt{2 π} σ} dt

6.18.53 NORMINV

Summary: returns the inverse of NORMDIST(x;Mean;StandardDeviation,TRUE()).

Syntax: NORMINV( Number P ; Number Mean ; Number StandardDeviation )

Constraints: StandardDeviation > 0 and 0 < P < 1.

Semantics: NORMINV returns the unique number x such that NORMDIST(x;Mean;StandardDeviation;TRUE()) = P.

6.18.54 LEGACY.NORMSDIST

Summary: returns the value of the cumulative distribution function for the standard normal distribution.

Syntax: LEGACY.NORMSDIST( Number X )

Semantics: LEGACY.NORMSDIST returns the value

\int_{- \infty}^{X} \frac{e^{- \frac{1}{2} \cdot t^{2}}}{\sqrt{2 π}} dt

This is exactly NORMDIST(X;0;1;TRUE()).

See also NORMDIST 6.18.52, LEGACY.NORMSINV 6.18.55

6.18.55 LEGACY.NORMSINV

Summary: returns the inverse of LEGACY.NORMSDIST(X).

Syntax: LEGACY.NORMSINV( Number P )

Semantics: LEGACY.NORMSINV returns NORMINV (P).

Constraints: 0 < P < 1.

See also NORMINV 6.18.53, LEGACY.NORMSDIST 6.18.54

6.18.56 PEARSON

Summary: PEARSON returns the Pearson correlation coefficient of two data sets

Syntax: PEARSON( ForceArray Array IndependentValues ; ForceArray Array DependentValues )

Constraints: COLUMNS(IndependentValues) = COLUMNS(DependentValues), ROWS(IndependentValues) = ROWS(DependentValues), both sequences shall contain at least one number at corresponding positions each.

For an empty element or an element of type Text or Boolean in IndependentValues the element at the corresponding position of DependentValues is ignored, and vice versa.

•IndependentValues: represents the array of the first data set. (X-Values)
•DependentValues: represents the array of the second data set. (Y-Values)

\begin{matrix} r = \frac{\sum_{i = 1}^{N} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2} \sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2}}} \\ \bar{x}, \bar{y} are the averages of the given x, y data \end{matrix}

See also COLUMNS 6.13.5, ROWS 6.13.30

6.18.57 PERCENTILE

Summary: Calculates the x-th sample percentile among the values in range.

Syntax: PERCENTILE( NumberSequenceList Data ; Number X )

Returns the X-th sample percentile of data values in Data. A percentile returns the scale value for a data series which goes from the smallest (Alpha = 0) to the largest value (Alpha = 1) of a data series. For Alpha = 25%, the percentile means the first quartile; Alpha = 50% is the MEDIAN.

•COUNT(Data) > 0
•0 ≤ X ≤ 1
•Semantics:
•Data: The array or range of values to get the percentile from.
•X: The percentile value between 0 and 1, inclusive. If X is not a multiple of $\frac{1}{n - 1}$
, PERCENTILE interpolates to obtain the value between two data points.
, PERCENTILE interpolates to obtain the value between two data points.

\begin{matrix} Step 1: \\ Sort the list of numbers given by array Data . \\ Step 2: \\ Calculate the ranking {1, \dots, n}, split into integer and decimal part \\ r = 1 + X \cdot (n - 1) = I + D \\ with \\ X = the percentile you want to find \\ n = the count of values \\ I = the integer part of the ranking = ⌊ r ⌋ \\ D = the decimal part of the ranking = r - ⌊ r ⌋ \\ Step 3: \\ Interpolate between the necessary two numbers \\ PERCENTILE = Y_{I} + D \cdot (Y_{I + 1} - Y_{I}) \\ with Y_{I} being the data point ranked at position I \end{matrix}

See also COUNT 6.13.6, MAX 6.18.45, MAX 6.18.45, MEDIAN 6.18.47, MIN 6.18.48, PERCENTRANK 6.18.58, QUARTILE 6.18.64, RANK 6.18.65

6.18.58 PERCENTRANK

Summary: Returns the percentage rank of a value in a sample.

Syntax: PERCENTRANK( NumberSequenceList Data ; Number X [ ; Integer Significance = 3 ] )

•COUNT(Data) > 0
•MIN(Data) ≤ X ≤ MAX(Data)
•INT(Significance) = Significance; Significance ≥ 1

Returns the rank of a value in a data set Data as a percentage of the data set, a value between 0 and 1, inclusive. This function can be used to evaluate the relative standing of a value within a data set.

•Data: the array or range of data with numeric values.
•X: the value whose rank is to be determined.
•Significance: an optional value that identifies the number of significant digits for the returned percentage value. If omitted, a value of 3 is used (0.xxx).

For COUNT(Data) > 1, PERCENTRANK returns r / (COUNT(Data) -1), where r is the rank of X in Data. The rank of the lowest number in Data is 0, and of the next lowest number 1, and so on. If X is not in Data, it is assigned a fractional rank proportionately between the rank of the numbers on either side. Specifically, if X lies between Y and Z = Y + 1 (Y < X < Z) with Y being the largest number smaller than X and Z the smallest number larger than X, and where Y has rank ry, the rank of X is calculated as

rx = ry + \frac{X - Y}{Z - Y}

In the special case where COUNT(Data) = 1, the only valid value for X is the single value in Data, in which case PERCENTRANK returns 1.

See also COUNT 6.13.6, INT 6.17.2, MAX 6.18.45, MIN 6.18.48, PERCENTILE 6.18.57, RANK 6.18.65

6.18.59 PERMUT

Summary: returns the number of permutations of k objects taken from n objects.

Syntax: PERMUT( Integer N ; Integer K )

Constraints: N ≥ 0; K ≥ 0; N ≥ K

Semantics: PERMUT returns

\frac{N!}{(N - K)!}

6.18.60 PERMUTATIONA

Summary: Returns the number of permutations for a given number of objects (repetition allowed).

Syntax: PERMUTATIONA( Integer Total ; Integer Chosen )

Constraints: Total ≥ 0, Chosen ≥ 0

Semantics: Given Total number of objects, return the number of permutations containing Chosen number of objects, with repetition permitted. The result is 1 if Total = 0 and Chosen = 0, otherwise the result is

PERMUTATIONA = {Total}^{Chosen}

6.18.61 PHI

Summary: Returns the values of the density function for a standard normal distribution.

Syntax: PHI( Number N )

Semantics: PHI(N) is a synonym for NORMDIST(N,0,1,FALSE()).

6.18.62 POISSON

Summary: returns the probability or the cumulative distribution function for the Poisson distribution

Syntax: POISSON( Integer X ; Number λ [ ; Logical Cumulative = TRUE ] )

Constraints: λ > 0, X ≥ 0

Semantics: If Cumulative is FALSE, POISSON returns the value

\frac{e^{- λ} λ^{⌊ X ⌋}}{⌊ X ⌋!}

If Cumulative is TRUE, POISSON returns the value

\sum_{k = 0}^{k = ⌊ X ⌋} \frac{e^{- λ} λ^{k}}{k!}

6.18.63 PROB

Summary: Returns the probability that a discrete random variable lies between two limits.

Syntax: PROB( ForceArray Array Data ; ForceArray Array Probability ; Number Start [ ; Number End ] )

•The sum of the probabilities in Probability shall equal 1.
•All values in Probability shall be > 0 and ≤ 1.
•COUNT(Data) = COUNT(Probability)

denotes the indicator function that is 1 if

•Data: the array or range of data in the sample ( the Number values in this array or range are referred to below as $d_{1}, d_{2}, \dots, d_{n}$
).
).
•Probability: the array or range of the corresponding probabilities ( the Number values in this array or range are referred to below as $p_{1}, p_{2}, \dots, p_{n}$
).
).
•Start: the start value (lower bound) of the interval whose probabilities are to be summed.
•End: (optional) the end value (upper bound) of the interval whose probabilities are to be summed. If omitted, End = Start is used.

Suppose that

I (x, a, b)

a \leq x \leq b

and 0 otherwise.

Then PROB returns

\sum_{i = 1}^{n} (I (d_{i}, Start, End) \times p_{i})

i.e. the sum of all probabilities

p_{i}

whose corresponding data value

d_{i}

satisfies

Start \leq d_{i} \leq End

Note that if

End < Start

then PROB returns 0 since in this case

I (d_{i}, Start, End) = 0

for all

6.18.64 QUARTILE

Summary: Returns a quartile of a set of data points.

Syntax: QUARTILE( NumberSequence Data ; Integer Quart )

•COUNT(Data) > 0
•0 ≤ Quart ≤ 4

Based on the statistical rank of the data points in Data, QUARTILE returns the percentile value indicated by Quart. The percentile is calculated as Quart divided by 4. An algorithm to calculate the percentile for a set of data points is given in the definition of PERCENTILE.

•Data: The cell range or data array of numeric values.
•Quart: The number of the quartile to return.
If Quart = 0, the minimum value is returned, which is equivalent to the MIN() function.
If Quart = 1, the value of the 25th percentile is returned.
If Quart = 2, the value of the 50th percentile is returned, which is equivalent to the MEDIAN() function.
If Quart = 3, the value of the 75th percentile is returned.
If Quart = 4, the maximum value is returned, which is equivalent to the MAX() function.

See also COUNT 6.13.6, MAX 6.18.45, MEDIAN 6.18.47, MIN 6.18.48, PERCENTILE 6.18.57, PERCENTRANK 6.18.58, RANK 6.18.65

6.18.65 RANK

Summary: Returns the rank of a number in a list of numbers.

Syntax: RANK( Number Value ; NumberSequenceList Data [ ; Number Order = 0 ] )

Constraints: Value shall exist in Data.

Semantics: The RANK function returns the rank of a value within a list.

•Value: the number for which to determine the rank.
•Data: numbers used to determine the ranking.
•Order: specifies how to rank the numbers:
If 0 or omitted, Data is ranked in descending order.
If not 0, Data is ranked in ascending order.

If a number in Data occurs more than once it is given the same rank, but increments the rank for subsequent different numbers. If Value does not exist in Data an Error is returned.

6.18.66 RSQ

Summary: Returns the square of the Pearson product moment correlation coefficient.

Syntax: RSQ( ForceArray Array ArrayY ; ForceArray Array ArrayX )

The arguments shall be either numbers or names, arrays, or references that contain numbers.

If an array or reference argument contains Text, Logical values, or empty cells, those values are ignored; however, cells with the value zero are included.

If ArrayY and ArrayX are empty or have a different number of data points, then #N/A is returned.

COLUMNS(ArrayY) = COLUMNS(ArrayX), ROWS(ArrayY) = ROWS(ArrayX)Semantics: The r-squared value can be interpreted as the proportion of the variance in y attributable to the variance in x.

\begin{matrix} r^{2} = \frac{\sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2} - \sum_{i = 1}^{N} {(y_{i} - y_{calc})}^{2}}{\sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2}} \\ y_{calc} = a + bx \\ and \\ a = \frac{(\sum_{i = 1}^{N} (x_{i}^{2})) (\sum_{i = 1}^{N} (y_{i})) - (\sum_{i = 1}^{N} (x_{i})) (\sum_{I = 1}^{N} (x_{i} y_{i}))}{N (\sum_{i = 1}^{N} (x_{i}^{2})) - {(\sum_{i = 1}^{N} (x_{i}))}^{2}} \\ b = \frac{N (\sum_{i = 1}^{N} (x_{i} y_{i})) - (\sum_{i = 1}^{N} (x_{i})) (\sum_{i = 1}^{N} (y_{i}))}{N (\sum_{i = 1}^{N} (x_{i}^{2})) - {(\sum_{i = 1}^{N} (x_{i}))}^{2}} \end{matrix}

The result of the RSQ function is the same as PEARSON * PEARSON.

For an empty element or an element of type Text or Boolean in ArrayY the element at the corresponding position of ArrayX is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30, PEARSON 6.18.56

6.18.67 SKEW

Summary: Estimates the skewness of a distribution using a sample set of numbers.

Syntax: SKEW( { NumberSequenceList Sample }+ )

Constraints: The sequence shall contain three numbers at least.

Semantics: Estimates the skewness of a distribution using a sample set of numbers.

and the standard deviation estimate

\bar{x}

s

, the skewness becomes

v = \frac{N}{(N - 1) (N - 2)} \sum_{i = 1}^{N} {(\frac{x_{i} - \bar{x}}{s})}^{3}

6.18.68 SKEWP

Summary: Calculates the skewness of a distribution using the population of a random variable.

Syntax: SKEWP( { NumberSequence Population }+ )

Constraints: The sequence shall contain three numbers at least.

Semantics: Calculates the skewness of a distribution using the population, i.e. the possible outcomes, of a random variable.

and the standard deviation

\bar{x}

,the skewness becomes

v = \frac{1}{N} \sum_{i = 1}^{N} {(\frac{x_{i} - \bar{x}}{σ})}^{3}

6.18.69 SLOPE

Summary: Calculates the slope of the linear regression line.

Syntax: SLOPE( ForceArray Array Y ; ForceArray Array X )

Constraints: COLUMNS(Y) = COLUMNS(X), ROWS(Y) = ROWS(X), both sequences shall contain at least one number at corresponding positions each.

Semantics: Calculates the slope of the linear regression line.

a = \frac{\sum_{i = 1}^{N} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2}}

For an empty element or an element of type Text or Boolean in Y the element at the corresponding position of X is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30, INTERCEPT 6.18.38, STEYX 6.18.76

6.18.70 SMALL

Summary: Finds the nth smallest value in a list.

Syntax: SMALL( NumberSequenceList List ; Integer|Array N )

Returns: Number or Array

Constraints: ROUNDDOWN(N;0) = N, effectively being INT(N) = N for positive numbers. If the resulting N is <1 or larger than the size of List, Error is returned.

Semantics: If N is an array of numbers, an array of smallest values is returned.

See also INT 6.17.2, LARGE 6.18.40, ROUNDDOWN 6.17.6

6.18.71 STANDARDIZE

Summary: Calculates a normalized value of a random variable.

Syntax: STANDARDIZE( Number Value ; Number Mean ; Number Sigma )

Semantics: Calculates a normalized value of a random variable.

Constraints: Sigma > 0

STANDARDIZE = \frac{(Value - Mean)}{Sigma}

6.18.72 STDEV

Summary: Compute the sample standard deviation of a set of numbers.

Syntax: STDEV( { NumberSequenceList N }+ )

Constraints: At least two numbers shall be included. Returns an Error if less than two Numbers are provided.

Semantics: Computes the sample standard deviation s, where

\begin{matrix} s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \\ with \\ \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} \end{matrix}

Note that s is not the same as the standard deviation of the set, σ, which uses n rather than n − 1.

6.18.73 STDEVA

Summary: Calculate the standard deviation using a sample set of values, including values of type Text and Logical.

Syntax: STDEVA( { Any Sample }+ )

Constraints: COUNTA(Sample) > 1.

Semantics: Unlike the STDEV function, includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1, and FALSE is treated as 0. Empty cells are not included.

The handling of string constants as parameters is implementation-defined. Either, string constants are converted to numbers, if possible and otherwise, they are treated as 0, or string constants are always treated as 0.

Suppose the resulting sequence of values is x1, x2, …, xn. Then let

\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}

STDEVA returns

s = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}

6.18.74 STDEVP

Summary: Calculates the standard deviation using the population of a random variable, including values of type Text and Logical.

Syntax: STDEVP( { NumberSequence N }+ )

Constraints: COUNT(N) ≥ 1.

Semantics: Computes the standard deviation of the set σ, where

σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}

Note that σ is not the same as the sample standard deviation, s, which uses n − 1 rather than n.

See also COUNT 6.13.6, STDEV 6.18.72, AVERAGE 6.18.3

6.18.75 STDEVPA

Summary: Calculates the standard deviation using the population of a random variable, including values of type Text and Logical.

Syntax: STDEVPA( { Any Sample }+ )

Constraints: COUNTA(Sample) ≥ 1.

the standard deviation becomes

\bar{x}

σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}

In the sequence, only Numbers and Logical types are considered; cells with Text are converted to 0; other types are ignored. If Logical types are a distinct type, they are still included, with TRUE considered 1 and FALSE considered 0. Any Sample may be of type ReferenceList.

The handling of string constants as parameters is implementation-defined. Either, string constants are converted to numbers, if possible and otherwise, they are treated as zero, or string constants are always treated as zero.

6.18.76 STEYX

Summary: Calculates the standard error of the predicted y value for each x in the regression.

Syntax: STEYX( ForceArray Array MeasuredY ; ForceArray Array X )

Constraints: COLUMNS(MeasuredY) = COLUMNS(X), ROWS(MeasuredY) = ROWS(X), both sequences shall contain at least three numbers at corresponding positions each.

Semantics: Calculates the standard error of the predicted y value for each x in the regression.

STEYX = \sqrt{\frac{1}{n (n - 2)} (n \sum_{n} y_{i}^{2} - {(\sum_{n} y_{i})}^{2} - \frac{{(n \sum_{n} x_{i} y_{i} - \sum_{n} x_{i} \sum_{n} y_{i})}^{2}}{n \sum_{n} x^{2} - {(\sum_{n} x)}^{2}})}

For an empty element or an element of type Text or Boolean in MeasuredY the element at the corresponding position of X is ignored, and vice versa.

See also COLUMNS 6.13.5, ROWS 6.13.30, INTERCEPT 6.18.38, SLOPE 6.18.69

6.18.77 LEGACY.TDIST

Summary: Returns the area to the tail or tails of the probability density function of the t-distribution.

Syntax: LEGACY.TDIST( Number X ; Integer Df ; Integer Tails)

Constraints: X ≥ 0, Df ≥ 1, Tails = 1 or 2

Semantics: Then LEGACY.TDIST returns

Tails \cdot \int_{x}^{\infty} f (t) dt

Note that Df denotes the degrees of freedom of the t-distribution and Γ is the Gamma function.

f (t) = \frac{Γ (\frac{Df + 1}{2})}{\sqrt{π Df} Γ (\frac{Df}{2})} {(1 + \frac{t^{2}}{Df})}^{- \frac{(Df + 1)}{2}}

See also GAMMA 6.16.34, BETADIST 6.18.7, BINOMDIST 6.18.10, CHISQDIST 6.18.12, EXPONDIST 6.18.21, FDIST 6.18.22, GAMMADIST 6.18.31, GAUSS 6.18.33, HYPGEOMDIST 6.18.37, LOGNORMDIST 6.18.44, NEGBINOMDIST 6.18.51, NORMDIST 6.18.52, POISSON 6.18.62, WEIBULL 6.18.86

6.18.78 TINV

Summary: Calculates the inverse of the two-tailed t-distribution.

Syntax: TINV( Number Probability ; Integer DegreeOfFreedom )

Constraints: 0 < Probability ≤ 1, DegreeOfFreedom ≥ 1

Semantics: Calculates the inverse of the two-tailed t-distribution.

6.18.79 TREND

Summary: Returns predicted values based on a simple or multiple linear regression.

Syntax: TREND( Array KnownY [ ; [ Array KnownX ] [ ; [ Array NewX ] [ ; Logical Const = TRUE ] ] ] )

KnownY: The set of known y-values to be used to determine the regression equation

KnownX: The set of known x-values to be used to determine the regression equation. If omitted or an empty parameter, it is set to the sequence 1,2,3,…,k, where k = ROWS(KnownY) ∙ COLUMNS(KnownY).

NewX: The set of x-values for which predicted y-values are to be calculated. If omitted or an empty parameter, it is set to KnownX.

Const: If set to FALSE, the model constant a is equal to 0.

LINEST(KnownY; KnownX; Const; FALSE()) either returns an error an array with 1 row and n + 1 columns. If it returns an error then so does TREND. If it returns an array, we call the entries in that array

b_{n}, b_{n - 1}, \dots, b_{1}, a

Let

z_{i j}

denote the entry in the ith row and jth column of NewX.

If COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = ROWS(KnownX), then TREND returns an array with ROWS(NewX) rows and COLUMNS(NewX) column, such that the entry in its ith row and jth column is

a + b_{1} \cdot z_{i j}

Otherwise, if COLUMNS(KnownY) = 1 and ROWS(KnownY) = ROWS(KnownX) and COLUMNS(KnownX) = COLUMNS(NewX), then TREND returns an array with ROWS(NewX) rows and 1 column, such that the entry in the ith row is

a + \sum_{j = 1}^{n} b_{j} \cdot z_{i j}

Otherwise, if COLUMNS(KnownY) = COLUMNS(KnownX) and ROWS(KnownY) = 1 and ROWS(KnownX) = ROWS(NewX), then TREND returns an array with 1 row and COLUMNS(NewX) columns, such that the entry in the jth column is

a + \sum_{i = 1}^{n} b_{i} \cdot z_{i j}

See also COLUMNS 6.13.5, ROWS 6.13.30, INTERCEPT 6.18.38, LINEST 6.18.41, SLOPE 6.18.69 , STEYX 6.18.76

6.18.80 TRIMMEAN

Summary: Returns the mean of a data set, ignoring a proportion of high and low values.

Syntax: TRIMMEAN( NumberSequenceList DataSet ; Number CutOffFraction )

Constraints: 0 ≤ CutOffFraction < 1

Semantics: Returns the mean of a data set, ignoring a proportion of high and low values.

Let n denote the number of elements in the data set and let

{SortedDataSet}_{1}, {SortedDataSet}_{2}, {SortedDataSet}_{3}, \dots, {SortedDataSet}_{n}

be the values in the data set sorted in ascending order. Moreover let

CutOff = INT (\frac{n \cdot CutOffFraction}{2})

Then TRIMMEAN returns the value

\frac{1}{n - 2 \cdot CutOff} \sum_{i = CutOff + 1}^{n - CutOff} {SortedDataSet}_{i}

See also AVERAGE 6.18.3 , GEOMEAN 6.18.34 , HARMEAN 6.18.36

6.18.81 TTEST

Summary: Calculates the p-value of a 2-sample t-test.

Syntax: TTEST( ForceArray Array X ; ForceArray Array Y ; Integer Tails ; Integer Type )

Constraints: COUNT(X) > 1, COUNT(Y) > 1, Tails = 1 or 2, Type = 1,2, or 3,
(COUNT(X) = COUNT(Y) or Type ≠ 1)

COLUMNS(X) = COLUMNS(Y), ROWS(X) = ROWS(Y)

Semantics: Let X1, X2, …,Xn be the numbers in the sequence X and Y1, Y2, …,Ym be the numbers in the sequence Y. Then

\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}

\bar{Y} = \frac{1}{m} \sum_{i = 1}^{m} Y_{i}

Moreover let

s_{X}^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {(X_{i} - \bar{X})}^{2}

s_{Y}^{2} = \frac{1}{m - 1} \sum_{i = 1}^{m} {(Y_{i} - \bar{Y})}^{2}

where Γ is the Gamma function.

f (x, df) = \frac{Γ (\frac{df + 1}{2})}{\sqrt{π df} Γ (\frac{df}{2})} {(1 + \frac{x^{2}}{df})}^{- \frac{(df + 1)}{2}}

(1)If type = 1, TTEST calculates the p-value for a paired-sample comparison of means test. Note that in this case due to the above constraints n = m. With
$s_{X - Y}^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {((X_{i} - Y_{i}) - (\bar{X} - \bar{Y}))}^{2}$
and
$t = | \frac{\bar{X} - \bar{Y}}{\sqrt{s_{X - Y}^{2}}} \sqrt{n} |$
TTEST returns
$Tails \cdot \int_{t}^{\infty} f (x, n - 1) dx$
(2)If Type = 2, TTEST calculates the p-value of a comparison of means for independent samples from populations with equal variance. With

(1) $s_{p}^{2} = \frac{(n - 1) s_{X}^{2} + (m - 1) s_{Y}^{2}}{n + m - 2}$

and
$t = | \frac{\bar{X} - \bar{Y}}{\sqrt{s_{p}^{2} (\frac{1}{n} + \frac{1}{m})}} |$

TTEST returns
$tails \cdot \int_{t}^{\infty} f (x, n + m - 2) dx$
(3)If Type = 3, TTEST calculates the p-value of a comparison of means for independent samples from populations with not necessarily equal variances. With

(2) $t = | \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{s_{X}^{2}}{n} + \frac{s_{Y}^{2}}{m}}} |$

and
$v = \frac{{(\frac{S_{X}^{2}}{n} + \frac{S_{Y}^{2}}{m})}^{2}}{\frac{{(\frac{S_{X}^{2}}{n})}^{2}}{(n - 1)} + \frac{{(\frac{S_{Y}^{2}}{m})}^{2}}{(m - 1)}}$

TTEST returns
$tails \cdot \int_{t}^{\infty} f (x, v) dx$

For an empty element or an element of type Text or Boolean in X the element at the corresponding position of Y is ignored, and vice versa.

See also COLUMNS 6.13.5, COUNT 6.13.6, ROWS 6.13.30, FTEST 6.18.30, LEGACY.TDIST 6.18.77, ZTEST 6.18.87

6.18.82 VAR

Summary: Compute the sample variance of a set of numbers.

Syntax: VAR( { NumberSequence N }+ )

Constraints: At least two numbers shall be included. Returns an Error if less than two Numbers are provided.

Semantics: Computes the sample variance s2, where

s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} = \frac{1}{n - 1} ((\sum_{i = 1}^{n} {x_{i}}^{2}) - n {\bar{x}}^{2})

Note that s2 is not the same as the variance of the set, σ2, which uses n rather than n − 1.

See also VARP 6.18.84, STDEV 6.18.72, AVERAGE 6.18.3

6.18.83 VARA

Summary: Estimates the variance using a sample set of values, including values of type Text and Logical.

Syntax: VARA( { Any Sample }+ )

Constraints: The sequence shall contain two numbers at least.

Semantics: Unlike the VAR function, includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1, and FALSE is treated as 0. Empty cells are not included.

the estimated variance becomes

\bar{x}

s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} = \frac{1}{n - 1} ((\sum_{i = 1}^{n} {x_{i}}^{2}) - n {\bar{x}}^{2})

6.18.84 VARP

Summary: Compute the variance of the set for a set of numbers.

Syntax: VARP( { NumberSequence N }+ )

Constraints: COUNT(N) ≥ 1

Semantics: Computes the variance of the set σ2, where

σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} = \frac{1}{n} ((\sum_{i = 1}^{n} {x_{i}}^{2}) - n {\bar{x}}^{2})

Note that σ2 is not the same as the sample variance, s2, which uses n − 1 rather than n.

If only one number is provided, returns 0.

See also COUNT 6.13.6, VAR 6.18.82, STDEVP 6.18.74, AVERAGE 6.18.3

6.18.85 VARPA

Summary: Calculates the variance using the population of the distribution, including values of type Text and Logical.

Syntax: VARPA( { Any Sample }+ )

Constraints: COUNTA(Sample) ≥ 1.

Semantics: Unlike the VARP function, includes values of type Text and Logical. Text values are treated as number 0. Logical TRUE is treated as 1, and FALSE is treated as 0. Empty cells are not included.

Summary: Calculates the Weibull distribution.

\bar{x}

the variance becomes

σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} = \frac{1}{n} ((\sum_{i = 1}^{n} {x_{i}}^{2}) - n {\bar{x}}^{2})

6.18.86 WEIBULL

Syntax: WEIBULL( Number Value ; Number Shape ; Number Scale ; Logical Cumulative )

Constraints: Value ≥ 0; Shape > 0; Scale > 0

Semantics: Calculates the Weibull distribution at the position Value.

If Cumulative is FALSE, the probability density function is calculated:

\frac{Shape}{Scale} {(\frac{Value}{Scale})}^{Shape - 1} e^{- {(\frac{Value}{Scale})}^{Shape}}

If Cumulative is TRUE, the cumulative distribution function is calculated:

1 - e^{- {(\frac{Value}{Scale})}^{Shape}}

See also BETADIST 6.18.7, BINOMDIST 6.18.10, CHISQDIST 6.18.12, EXPONDIST 6.18.21, FDIST 6.18.22, GAMMADIST 6.18.31, GAUSS 6.18.33, HYPGEOMDIST 6.18.37, LOGNORMDIST 6.18.44, NEGBINOMDIST 6.18.51, NORMDIST 6.18.52, POISSON 6.18.62, LEGACY.TDIST 6.18.77

6.18.87 ZTEST

Summary: Calculates the probability of observing a sample mean as large or larger than the mean of the given sample for samples drawn from a normal distribution.

Syntax: ZTEST( NumberSequenceList Sample ; Number Mean [ ; Number Sigma ] )

Constraints: The sequence Sample shall contain at least two numbers.

Semantics: Calculates the probability of observing a sample mean as large or larger than the mean of the given Sample for samples drawn from a normal distribution with the given mean Mean and the given standard deviation Sigma. If Sigma is omitted, it is estimated from Sample, using STDEV. With Sample being the mean of Sample and

z = \frac{\bar{Sample} - Mean}{Sigma} \sqrt{n}

ZTEST returns

P (z \leq Z) = \frac{1}{\sqrt{2 π}} \int_{z}^{\infty} e^{- \frac{x^{2}}{2}} dx

6.19 Number Representation Conversion Functions

6.19.1 General

These functions convert between different representations of numbers, such as between different bases and Roman numerals.

The base conversion functions xxx2BIN (such as DEC2BIN), xxx2OCT, and xxx2HEX functions return Text, while the xxx2DEC functions return Number. All of the xxx2yyy functions accept either Text or Number, though a Number is interpreted as the digits when printed in base 10. These are intended to support relatively small numbers, and have a somewhat convoluted interface and semantics, as described in their specifications. General base conversion capabilities are provided by BASE and DECIMAL.

As an argument for the HEX2xxx functions, a hexadecimal number is any string consisting solely of the characters "0","1" to "9", "a" to "f" and "A" to "F". The hexadecimal output of an xxx2HEX function shall be a string consisting solely of the characters "0","1" to "9" (U+0030 through U+0039), "a" to "f" (U+0061 through U+0066) and "A" to "F" (U+0041 through U+0046), and should be a string consisting solely of the characters "0","1" to "9" and "A" to "F". In both cases, the 40th bit (from the right) is considered a sign bit.

6.19.2 ARABIC

Summary: Convert Roman numerals to Number.

Syntax: ARABIC( Text X )

Constraints: X shall contain Roman numerals, or an empty string.

Semantics: Converts the Roman numeral to Number. This is the reverse of ROMAN; see ROMAN for the values of individual Roman numeral symbols. A Roman symbol to the left of a larger symbol (directly or indirectly) reduces the final value by the symbol amount, otherwise, it increases the final amount by the symbol's amount. Case is ignored.

The characters accepted are U+004D "M", U+0044 "D", U+0043 "C", U+004C "L", U+0058 "X", U+0056 "V", U+0049 "I", U+006D "m", U+0064 "d", U+0063 "c", U+006C "l", U+0078 "x", U+0076 "v", U+0069 "i" .

The following identity shall hold: ARABIC(ROMAN(x; any)) = x, when ROMAN(x; any) is not an Error.

If X is an empty string, 0 is returned.

6.19.3 BASE

Summary: Converts a number into a text representation with the given base.

Syntax: BASE( Integer X ; Integer Radix [ ; Integer MinimumLength ] )

Constraints: X ≥ 0, 2 ≤ Radix ≤ 36, MinimumLength ≥ 0

Semantics: Converts number X into text that represents the value of X in base Radix. The symbols 0-9 (U+0030 through U+0039), then upper case A-Z (U+0041 through U+005A) are used as digits. Thus, BASE(45745;36) returns “ZAP”.

If MinimumLength is not supplied, the generated text uses the smallest number of characters (i.e., it does not add leading 0s). If MinimumLength is supplied, and the resulting text would normally be smaller than MinimumLength, leading 0s are added to produce text exactly MinimumLength characters long. If the text is longer than the MinimumLength argument, the MinimumLength parameter is ignored.

6.19.4 BIN2DEC

Summary: Converts a binary (base 2) number (up to 10 digits) to its decimal equivalent

Syntax: BIN2DEC( TextOrNumber X )

Constraints: X shall contain only binary digits (no space or other characters), and shall contain at least one binary digit. When considered as a Number, INT(X) = X. Evaluators may evaluate expressions where the digits in X are only 0 or 1, no more than 10 digits.

Semantics: Converts given binary number into decimal equivalent, with the topmost 10th digit being the sign bit (using a two's-complement representation). If given Text, the text is considered a binary number representation. If given a Number, the digits of the number when printed as base 10 are considered the digits of the equivalently-represented binary number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

If any digits are 2 through 9, an evaluator shall return an Error. It is implementation-defined what happens if an evaluator is given an empty string; evaluators may return an Error or 0 in such cases.

Summary: Converts a binary (base 2) number (10th bit is sign) to its hexadecimal equivalent

6.19.5 BIN2HEX

Syntax: BIN2HEX( TextOrNumber X [ ; Number Digits ] )

Semantics: Converts given binary number into hexadecimal (base 16) equivalent. For input value X, the topmost 10th digit is considered the sign bit (using a two's-complement representation). If given Text, the text is considered a binary number representation. If given a Number, the digits of the number when printed as base 10 are considered the digits of the equivalently-represented binary number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

If any digits in X are 2 through 9, an evaluator shall return an Error. It is implementation-defined what happens if an evaluator is given an empty string; evaluators may return an Error or 0 in such cases.

The resulting value is a hexadecimal value, up to 10 hexadecimal digits, with the topmost bit (40th bit) being the sign bit and in two's-complement form. The digits A through F are in uppercase. If the input has its 10th bit on, the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits required than the Digits parameter specifies, the results are implementation-defined.

Summary: Converts a binary (base 2) number (10th bit is sign) to its octal (base 8) equivalent

6.19.6 BIN2OCT

Syntax: BIN2OCT( TextOrNumber X [ ; Number Digits ] )

Semantics: Converts given binary number into octal (base 8) equivalent. For input value X, the topmost 10th digit is considered the sign bit (using a two's-complement representation). If given Text, the text is considered a binary number representation. If given a Number, the digits of the number when printed as base 10 are considered the digits of the equivalently-represented binary number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is an octal value, up to 10 octal digits, with the topmost bit (30th bit) being the sign bit and in two's-complement form. If the input has its 10th bit on, the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, its results are implementation-defined.

Summary: Converts a decimal number to base 2 (whose 10th bit is sign)

6.19.7 DEC2BIN

Syntax: DEC2BIN( TextOrNumber X [ ; Number Digits ] )

Constraints: X shall contain only decimal digits (no space or other characters), and shall contain at least one decimal digit. When considered as a Number, INT(X) = X. Evaluators may evaluate expressions where -512 ≤ X ≤ 511.

Semantics: Converts given number into binary (base 2) equivalent. If given Text, the text is considered a decimal number representation, and may have a leading minus sign. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a binary value, up to 10 digits, with the topmost bit (10th bit) being the sign bit and in two's-complement form. If the input is negative, the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Summary: Converts a decimal number to base 16 (whose 40th bit is sign)

6.19.8 DEC2HEX

Syntax: DEC2HEX( TextOrNumber X [ ; Number Digits ] )

Semantics: Converts given number into hexadecimal (base 16) equivalent. If given Text, the text is considered a decimal number representation, and may have a leading minus sign. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a hexadecimal value, up to 10 digits, with the topmost bit (40th bit) being the sign bit and in two's-complement form. If the input is negative, the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Summary: Converts a decimal number to base 8 (whose 30th bit is sign)

6.19.9 DEC2OCT

Syntax: DEC2OCT( TextOrNumber X [ ; Number Digits ] )

Semantics: Converts given number into octal (base 8) equivalent. If given Text, the text is considered a decimal number representation, and may have a leading minus sign. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a octal value, up to 10 digits, with the topmost bit (30th bit) being the sign bit and in two's-complement form. If the input is negative, the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

6.19.10 DECIMAL

Summary: Converts text representing a number in a given base into a base 10 number.

Syntax: DECIMAL( Text X ; Integer Radix )

Constraints: 2 ≤ Radix ≤ 36

Semantics: Converts text X in base Radix to a Number. Uppercase letters (U+0041 through U+005A) and lowercase letters (U+0061 through U+007A) are both accepted as equivalent if Radix > 10. Thus, DECIMAL("zap";36) and DECIMAL("ZAP";36) both compute 45745.

An Error is returned if X has characters that do not belong in base Radix. However, leading spaces and tabs in X are always ignored. If Radix is 16, a leading regular expression “0?[Xx]” is ignored, as is a trailing letter H or h. If Radix is 2, the letter b or B at the end is ignored (if present).

6.19.11 HEX2BIN

Summary: Converts a hexadecimal number (40th bit is sign) to base 2 (whose 10th bit is sign)

Syntax: HEX2BIN( TextOrNumber X [ ; Number Digits ] )

Constraints: X shall contain only hexadecimal digits (no space or other characters), and shall contain at least one hexadecimal digit. When considered as a Number, INT(X) = X. Evaluators may evaluate expressions where X is considered in base 10, -512 ≤ X ≤ 511.

Semantics: Converts given hexadecimal number into binary (base 2) equivalent. If given Text, the text is considered a hexadecimal number representation; if its 40th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a binary value, up to 10 digits, with the topmost bit (10th bit) being the sign bit and in two's-complement form. If the input is negative (40th bit is 1), the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Summary: Converts a hexadecimal number (40th bit is sign) to decimal

6.19.12 HEX2DEC

Syntax: HEX2DEC( TextOrNumber X )

Semantics: Converts given hexadecimal number into decimal equivalent. If given Text, the text is considered a hexadecimal number representation. If X's 40th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a decimal number.

Summary: Converts a hexadecimal number (40th bit is sign) to base 8 (whose 30th bit is sign)

6.19.13 HEX2OCT

Syntax: HEX2OCT( TextOrNumber X [ ; Number Digits ] )

Constraints: X shall contain hexadecimal digits (no spaces or other characters), and shall contain at least one hexadecimal digit. When considered as Number, INT(X) = X. Evaluators may evaluate expressions where X has 1 to 10 (inclusive) hexadecimal digits, base 10 value of X is -2 29 < X < 2 29 -1.

Semantics: Converts given hexadecimal number into octal (base 8) equivalent. If given Text, the text is considered a hexadecimal number representation; if its 40th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is an octal value, up to 10 digits, with the topmost bit (10th bit) being the sign bit and in two's-complement form. If the input is negative (40th bit is 1), the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Summary: Converts an octal number (30th bit is sign) to base 2 (whose 10th bit is sign)

6.19.14 OCT2BIN

Syntax: OCT2BIN( TextOrNumber X [ ; Number Digits ] )

Constraints: X shall contain only octal digits (no space or other characters), and shall contain at least one octal digit. When considered as a Number, INT(X) = X. Evaluators may evaluate expressions where X is considered in base 10, -512 ≤ X ≤ 511.

Semantics: Converts given octal (base 8) number into binary (base 2) equivalent. If given Text, the text is considered an octal number representation; if its 30th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a binary value, up to 10 digits, with the topmost bit (10th bit) being the sign bit and in two's-complement form. If the input is negative (30th bit is 1), the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Syntax: OCT2DEC( TextOrNumber X )

6.19.15 OCT2DEC

Summary: Converts an octal number (30th bit is sign) to decimal

Semantics: Converts given octal number into decimal equivalent. If given Text, the text is considered a octal number representation. If X's 30th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a decimal number.

Summary: Converts an octal number (30th bit is sign) to hexadecimal (whose 40th bit is sign)

6.19.16 OCT2HEX

Syntax: OCT2HEX( TextOrNumber X [ ; Number Digits ] )

Semantics: Converts given octal (base 8) number into hexadecimal (base 16) equivalent. If given Text, the text is considered an octal number representation; if its 30th bit is 1, it is considered a negative number. It is implementation-defined what happens if given a Logical value; an evaluator may produce an Error, or it may convert the Logical to Number (per “Convert to Number”) and then process as a Number.

The resulting value is a hexadecimal value, up to 10 digits, with the topmost bit (40th bit) being the sign bit and in two's-complement form. If the input is negative (30th bit is 1), the Digits argument is ignored; otherwise, the Digits indicates the number of digits in the output, with leading 0 digits added as necessary to bring it up to that number of digits. If there are more digits than specified by the Digits parameter, the results are implementation-defined.

Summary: Convert to Roman numerals

6.19.17 ROMAN

Syntax: ROMAN( Integer N [ ; Integer Format = 0 ] )

Constraints: N ≥ 0, N < 4000, 0 ≤ Format ≤ 4, ISLOGICAL(1) or NOT(ISLOGICAL(Format))

Semantics: Return the Roman numeral representation of N. Format specifies the level of conciseness, and defaults to 0, the classic representation, with larger numbers requiring increasing conciseness.

To support legacy documents, evaluators with Logical types that are distinct from Number may accept the format parameter as a scalar, and accept TRUE specifying format 0, and FALSE specifying format 4.

The following identity shall hold: ARABIC(ROMAN(x; any)) = x, when ROMAN(x; any) is not an Error.

If N is 0, an empty string is returned.

Table 31 - ROMAN lists the values of individual roman numerals; roman numerals that precede (directly or indirectly) a larger-valued roman number subtract their value from the final value.

Table 31 - ROMAN

Roman Numeral	Value	Unicode Code Point
I	1	U+0049
V	5	U+0056
X	10	U+0058
L	50	U+004C
C	100	U+0043
D	500	U+0044
M	1000	U+004D

Evaluators that accept 0 as a value of N should return the string “0”. Evaluators that accept negative values of N should include a negative sign (“-”) as the first character.

The Format levels are:

Table 32 - ROMAN

Format	Meaning
0 or omitted (or TRUE)	Only subtract powers of 10, not L or V, and only if the next number is not more than 10 times greater. A number following the larger one shall be smaller than the subtracted number. Also known as classic.
1	Powers of 10, and L and V may be subtracted, only if the next number is not more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
2	Powers of 10 and L, but not V, may be subtracted, also if the next number is more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
3	Powers of 10, and L and V may be subtracted, also if the next number is more than 10 times greater. A number following the larger one shall be smaller than the subtracted number.
4 (or FALSE)	Produce the fewest Roman digits possible. Also known as simplified.

See also Infix Operator "&" 6.4.10, ISLOGICAL 6.13.19, ARABIC 6.19.2

6.20 Text Functions

6.20.1 General

6.20.2 ASC

Summary: Converts full-width to half-width ASCII and katakana characters.

Syntax: ASC( Text T )

Summary: Return character represented by the given numeric value

Semantics: Conversion is done for full-width ASCII and [UNICODE] katakana characters, some characters are converted in a special way, see table below. Other characters are copied from T to the result. This is the complementary function to JIS.

The percent sign % in the conversion table below denotes the modulo operation. A followed by means that a character is converted to two consecutive characters.

Table 33 - ASC

From Unicode Character (c)	To Unicode Character	Comment
U+30a1 ≤ c ≤ U+30aa if c%2==0	(c - 0x30a2) / 2 + 0xff71	katakana a-o
U+30a1 ≤ c ≤ U+30aa if c%2==1	(c - 0x30a1) / 2 + 0xff67	katakana small a-o
U+30ab ≤ c ≤ U+30c2 if c%2==1	(c - 0x30ab) / 2 + 0xff76	katakana ka-chi
U+30ab ≤ c ≤ U+30c2 if c%2==0	(c - 0x30ac) / 2 + 0xff76 followed by 0xff9e	katakana ga-dhi
U+30c3	0xff6f	katakana small tsu
U+30c4 ≤ c ≤ U+30c9 if c%2==0	(c - 0x30c4) / 2 + 0xff82	katakana tsu-to
U+30c4 ≤ c ≤ U+30c9 if c%2==1	(c - 0x30c5) / 2 + 0xff82 followed by 0xff9e	katakana du-do
U+30ca ≤ c ≤ U+30ce	c - 0x30ca + 0xff85	katakana na-no
U+30cf ≤ c ≤ U+30dd if c%3==0	(c - 0x30cf) / 3 + 0xff8a	katakana ha-ho
U+30cf ≤ c ≤ U+30dd if c%3==1	(c - 0x30d0) / 3 + 0xff8a followed by 0xff9e	katakana ba-bo
U+30cf ≤ c ≤ U+30dd if c%3==2	(c - 0x30d1) / 3 + 0xff8a followed by 0xff9f	katakana pa-po
U+30de ≤ c ≤ U+30e2	c - 0x30de + 0xff8f	katakana ma-mo
U+30e3 ≤ c ≤ U+30e8 if c%2==0	(c - 0x30e4) / 2 + 0xff94)	katakana ya-yo
U+30e3 ≤ c ≤ U+30e8 if c%2==1	(c - 0x30e3) / 2 + 0xff6c	katakana small ya-yo
U+30e9 ≤ c ≤ U+30ed	c - 0x30e9 + 0xff97	katakana ra-ro
U+30ef	U+ff9c	katakana wa
U+30f2	U+ff66	katakana wo
U+30f3	U+ff9d	katakana nn
U+ff01 <= c <= U+ff5e	c - 0xff01 + 0x0021	ASCII characters
U+2015	U+ff70	HORIZONTAL BAR => HALFWIDTH KATAKANA-HIRAGANA PROLONGED SOUND MARK
U+2018	U+0060	LEFT SINGLE QUOTATION MARK => GRAVE ACCENT
U+2019	U+0027	RIGHT SINGLE QUOTATION MARK => APOSTROPHE
U+201d	U+0022	RIGHT DOUBLE QUOTATION MARK => QUOTATION MARK
U+3001	U+ff64	IDEOGRAPHIC COMMA
U+3002	U+ff61	IDEOGRAPHIC FULL STOP
U+300c	U+ff62	LEFT CORNER BRACKET
U+300d	U+ff63	RIGHT CORNER BRACKET
U+309b	U+ff9e	KATAKANA-HIRAGANA VOICED SOUND MARK
U+309c	U+ff9f	KATAKANA-HIRAGANA SEMI-VOICED SOUND MARK
U+30fb	U+ff65	KATAKANA MIDDLE DOT
U+30fc	U+ff70	KATAKANA-HIRAGANA PROLONGED SOUND MARK
U+ffe5	U+005c	FULLWIDTH YEN SIGN => REVERSE SOLIDUS "\"
Note 1: The "\" (REVERSE SOLIDUS, U+005C) is a specialty that gets displayed as a Yen sign with some Japanese fonts, which is a legacy of code-page 932.

Note 2: For references regarding halfwidth and fullwidth characters see [UAX11] and the Halfwidth and Fullwidth Code Chart of [UNICODE].

Note 3: For information about the mapping of JIS X 0201 and JIS X 0208 to Unicode characters see [JISX0201] and [JISX0208].

6.20.3 CHAR

Syntax: CHAR( Number N )

Constraints: N ≤ 127; Evaluators may evaluate expressions where N ≥ 1, N ≤ 255.

Returns character represented by the given numeric value.

Evaluators should return an Error if N > 255.

Evaluators should implement CHAR such that CODE(CHAR(N)) returns N for any 1 ≤ N ≤ 255.

Note 1: Beyond 127, some evaluators return a character from a system-specific code page, while others return the [UNICODE] character. Most evaluators do not allow values greater than 255.

Note 2: Where interoperability is a concern, expressions should use the UNICHAR function. 6.20.25

See also CODE 6.20.5, UNICHAR 6.20.25, UNICODE 6.20.26

6.20.4 CLEAN

Summary: Remove all non-printable characters from the string and return the result.

Syntax: CLEAN( Text T )

Summary: Return numeric value corresponding to the first character of the text value.

Removes all non-printable characters from the string T and returns the resulting string. Evaluators should remove each particular character from the string, if and only if the character belongs to [UNICODE] class Cc (Other - Control), or to Unicode class Cn (Other - Not Assigned). The resulting string shall contain all printable characters from the original string, in the same order. The space character is considered a printable character.

6.20.5 CODE

Syntax: CODE( Text T )

Constraints: code point ≤ 127. Evaluators may evaluate expressions where Length(T) > 0.

Returns a numeric value which represents the first letter of the given text T.

Behavior for code points ≥ 128 is implementation-defined. Evaluators may use the underlying system's code page. Evaluators should implement CODE such that CODE(CHAR(N)) returns N for 1 ≤ N ≤ 255.

Note: Where interoperability is a concern, expressions should use the UNICODE function. 6.20.26

See also CHAR 6.20.3, UNICHAR 6.20.25, UNICODE 6.20.26

6.20.6 CONCATENATE

Summary: Concatenate the text strings

Syntax: CONCATENATE( { Text T }+ )

Semantics: Concatenate each text value, in order, into a single text result.

6.20.7 DOLLAR

Summary: Convert the parameters to Text formatted as currency.

Syntax: DOLLAR( Number N [ ; Integer D ] )

Semantics: Returns the value formatted as a currency, using locale-specific data. D is the number of decimal places used in the result string, a negative D rounds number N. If D is omitted, locale information may be used to determine the currency's decimal places, or a value of 2 shall be assumed.

6.20.8 EXACT

Summary: Report if two text values are equal using a case-sensitive comparison .

Syntax: EXACT( Text T1 ; Text T2 )

Semantics: Converts both sides to Text, and then returns TRUE if the two text values are equal, including case, otherwise it returns FALSE.

See also FIND 6.20.9, SEARCH 6.20.20, Infix Operator "<>" 6.4.8, Infix Operator "=" 6.4.7

6.20.9 FIND

Summary: Return the starting position of a given text.

Syntax: FIND( Text Search ; Text T [ ; Integer Start = 1 ] )

Semantics: Returns the character position where Search is first found in T, when the search is started from character position Start. The match is case-sensitive, and no wildcards or other instructions are considered in Search. Returns an Error if text not found.

Constraints: Start ≥ 1

6.20.10 FIXED

Summary: Round the number to a specified number of decimals and format the result as a text.

Syntax: FIXED( Number N [ ; Integer D = 2 [ ; Logical OmitSeparators = FALSE ] ] )

Semantics: Rounds value N to D decimal places (after the decimal point) and returns the result formatted as text, using locale-specific settings. If D is negative, the number is rounded to ABS(D) places to the left from the decimal point. If the optional parameter OmitSeparators is TRUE, then group separators are omitted from the resulting string. Group separators are included in the absence of this parameter. If D is a fraction, it is rounded towards 0 as an integer (ignoring what is the closest integer).

6.20.11 JIS

Summary: Converts half-width to full-width ASCII and katakana characters.

Syntax: JIS( Text T )

Summary: Return a selected number of text characters from the left.

Semantics: Conversion is done for half-width ASCII and [UNICODE] katakana characters, some characters are converted in a special way, see table below. Other characters are copied from T to the result. This is the complementary function to ASC.

A followed by means that there are two consecutive characters to convert from.

Table 34 - JIS

From Unicode Character (c)	To Unicode Character	Comment
U+0022	0x201d	QUOTATION MARK => RIGHT DOUBLE QUOTATION MARK This is an exception to the ASCII range that follows below.
U+005c	0xffe5	REVERSE SOLIDUS "\" => FULLWIDTH YEN SIGN (code-page 932 legacy, for details see ASC function) This is an exception to the ASCII range that follows below.
U+0060	0x2018	GRAVE ACCENT => LEFT SINGLE QUOTATION MARK This is an exception to the ASCII range that follows below.
U+0027	0x2019	APOSTROPHE => RIGHT SINGLE QUOTATION MARK This is an exception to the ASCII range that follows below.
U+0021 ≤ c ≤= U+007e	c - 0x0021 + 0xff01	ASCII characters
U+ff66	0x30f2	katakana wo
U+ff67 ≤ c ≤ U+ff6b	(c - 0xff67) * 2 + 0x30a1	katakana small a-o
U+ff6c ≤ c ≤ U+ff6e	(c - 0xff6c) * 2 + 0x30e3	katakana small ya-yo
U+ff6f	0x30c3	katakana small tsu
U+ff71 ≤ c ≤ U+ff75	(c - 0xff71) * 2 + 0x30a2	katakana a-o
U+ff76 ≤ c ≤ U+ff81 followed by U+ff9e	(c - 0xff76) * 2 + 0x30ac	katakana ga-dsu
U+ff76 ≤ c ≤ U+ff81 not followed by U+ff9e	(c - 0xff76) * 2 + 0x30ab	katakana ka-chi
U+ff82 ≤ c ≤ U+ff84 followed by U+ff9e	(c - 0xff82) * 2 + 0x30c5	katakana du-do
U+ff82 ≤ c ≤ U+ff84 not followed by U+ff9e	(c - 0xff82) * 2 + 0x30c4	katakana tsu-to
U+ff85 ≤ c ≤ U+ff89	c - 0xff85 + 0x30ca	katakana na-no
U+ff8a ≤ c ≤ U+ff8e followed by U+ff9e	(c - 0xff8a) * 3 + 0x30d0	katakana ba-bo
U+ff8a ≤ c ≤ U+ff8e followed by U+ff9f	(c - 0xff8a) * 3 + 0x30d1	katakana pa-po
U+ff8a ≤ c ≤ U+ff8e neither followed by U+ff9e nor U+ff9f	(c - 0xff8a) * 3 + 0x30cf	katakana ha-ho
U+ff8f ≤ c ≤ U+ff93	c - 0xff8f + 0x30de	katakana ma-mo
U+ff94 ≤ c ≤ U+ff96	(c - 0xff94) * 2 + 0x30e4	katakana ya-yo
U+ff97 ≤ c ≤ U+ff9b	c - 0xff97 + 0x30e9	katakana ra-ro
U+ff9c	U+30ef	katakana wa
U+ff9d	U+30f3	katakana nn
U+ff9e	U+309b	HALFWIDTH KATAKANA VOICED SOUND MARK => FULLWIDTH
U+ff9f	U+309c	HALFWIDTH KATAKANA SEMI-VOICED SOUND MARK => FULLWIDTH
U+ff70	U+30fc	HALFWIDTH KATAKANA-HIRAGANA PROLONGED SOUND MARK => FULLWIDTH
U+ff61	U+3002	HALFWIDTH IDEOGRAPHIC FULL STOP => FULLWIDTH
U+ff62	U+300c	HALFWIDTH LEFT CORNER BRACKET => FULLWIDTH
U+ff63	U+300d	HALFWIDTH RIGHT CORNER BRACKET => FULLWIDTH
U+ff64	U+3001	HALFWIDTH IDEOGRAPHIC COMMA => FULLWIDTH
U+ff65	U+30fb	HALFWIDTH KATAKANA MIDDLE DOT => FULLWIDTH

Note 1: For references regarding halfwidth and fullwidth characters see [UAX11] and the Halfwidth and Fullwidth Code Chart of [UNICODE].

Note 2: For information about the mapping of JIS X 0201 and JIS X 0208 to Unicode characters see [JISX0201] and [JISX0208].

6.20.12 LEFT

Syntax: LEFT( Text T [ ; Integer Length ] )

Semantics: Returns the INT(Length) number of characters of text T, starting from the left. If Length is omitted, it defaults to 1; otherwise, it computes Length = INT(Length). If T has fewer than Length characters, it returns T. This means that if T is an empty string (which has length 0) or the parameter Length is 0, LEFT() will always return an empty string. Note that if Length < 0, an Error is returned. This function shall return the same string as MID(T; 1; Length).

Constraints: Length ≥ 0

The results of this function may be normalization-sensitive. 4.2

See also INT 6.17.2, LEN 6.20.13, MID 6.20.15, RIGHT 6.20.19

6.20.13 LEN

Summary: Return the length, in characters, of given text

Syntax: LEN( Text T )

Returns: Integer

Semantics: Computes number of characters (not the number of bytes) in T. If T is of type Number, it is automatically converted to Text, including a fractional part and decimal separator if necessary.

The results of this function may be normalization-sensitive. 4.2

See also TEXT 6.20.23, ISTEXT 6.13.25, LEFT 6.20.12, MID 6.20.15, RIGHT 6.20.19

6.20.14 LOWER

Summary: Return input string, but with all uppercase letters converted to lowercase letters.

Syntax: LOWER( Text T )

Summary: Returns extracted text, given an original text, starting position, and length.

Semantics: Return input string, but with all uppercase letters converted to lowercase letters, as defined by §3.13 Default Case Algorithms, §4.2 Case-Normative and §5.18 Case Mappings of [UNICODE]. As with most functions, it is side-effect free (it does not modify the source values). All Evaluators shall convert A-Z to a-z.

Note: As this function can be locale aware, results may be unexpected in certain cases. For example in a Turkish locale an upper case "I without dot" (LATIN CAPITAL LETTER I, U+0049) is converted to a lower case "i without dot" (LATIN SMALL LETTER DOTLESS I, U+0131).

6.20.15 MID

Syntax: MID( Text T ; Integer Start ; Integer Length )

Constraints: Start ≥ 1, Length ≥ 0.

Semantics: Returns the characters from T, starting at character position Start, for up to Length characters. For the integer conversions, Start = INT(Start), and Length = INT(Length). If there are less than Length characters starting at start, it returns as many characters as it can beginning with Start. In particular, if Start > LEN(T), it returns the empty string (""). If Start < 0, it returns an Error. If Start ≥ 0, and Length = 0, it returns the empty string. Note that MID(T;1;Length) produces the same results as LEFT(T;Length).

The results of this function may be normalization-sensitive. 4.2

See also INT 6.17.2, LEFT 6.20.12, LEN 6.20.13, RIGHT 6.20.19, REPLACE 6.20.17, SUBSTITUTE 6.20.21

6.20.16 PROPER

Summary: Return the input string with the first letter of each word converted to an uppercase letter and the rest of the letters in the word converted to lowercase.

Syntax: PROPER( Text T )

Semantics: Return input string, but modified as follows:

●If the first character is a letter, it is converted to its uppercase equivalent; otherwise, the original character is returned
●If a letter is preceded by a non-letter, it is converted to its uppercase equivalent
●If a letter is preceded by a letter, it is converted to its lowercase equivalent.

Evaluators shall implement this for at least the Latin letters A-Z and a-z.

As with most functions, it is side-effect free, that is, it does not modify the source values.

6.20.17 REPLACE

Summary: Returns text where an old text is substituted with a new text.

Syntax: REPLACE( Text T ; Number Start ; Number Count ; Text New )

Semantics: Returns text T, but remove the characters starting at character position Start for Count characters, and instead replace them with New. Character positions defined by Start begin at 1 (for the leftmost character). If Count=0, the text New is inserted before character position Start, and all the text before and after Start is retained. If Start > length of text T (TLen) then Start is set to TLen. If Count > TLen - Start then Count is set to TLen - Start.

Constraints: Start ≥ 1.

REPLACE(T;Start;Len;New) is the same as LEFT(T;Start - 1) & New & MID(T; Start + Len; LEN(T)))

See also LEFT 6.20.12, LEN 6.20.13, MID 6.20.15, RIGHT 6.20.19, SUBSTITUTE 6.20.21

6.20.18 REPT

Summary: Return text repeated Count times.

Syntax: T( Text T ; Integer Count )

Semantics: Returns text T repeated Count number of times; if Count is zero, an empty string is returned. If Count < 0, the result is Error.

Constraints: Count ≥ 0

See also LEFT 6.20.12, MID 6.20.15, RIGHT 6.20.19, SUBSTITUTE 6.20.21

6.20.19 RIGHT

Summary: Return a selected number of text characters from the right.

Syntax: RIGHT( Text T [ ; Integer Length ] )

Semantics: Returns the Length number of characters of text T, starting from the right. If Length is omitted, it defaults to 1; otherwise, it computes Length = INT(Length). If T has fewer than Length characters, it returns T (unchanged). This means that if T is an empty string (which has length 0) or the parameter Length is 0, RIGHT() will always return an empty string. Note that if Length < 0, an Error is returned.

Constraints: Length ≥ 0

The results of this function may be normalization-sensitive. 4.2

See also INT 6.17.2, LEFT 6.20.12, LEN 6.20.13, MID 6.20.15

6.20.20 SEARCH

Summary: Return the starting position of a given text.

Syntax: SEARCH( Text Search ; Text T [ ; Integer Start = 1 ] )

Returns: Integer

Constraints: Start ≥ 1

Semantics: Returns the character position where Search is first found in T, when the search is started from character position Start. The match is not case-sensitive. Start is 1 if omitted. Returns an Error if text not found.

The values returned may vary depending upon the HOST-USE-REGULAR-EXPRESSIONS or HOST-USE-WILDCARDS properties. 3.4

6.20.21 SUBSTITUTE

Summary: Returns text where an old text is substituted with a new text.

Syntax: SUBSTITUTE( Text T ; Text Old ; Text New [ ; Integer Which ] )

Constraints: Which ≥ 1 (when provided)

Semantics: Returns text T, but with text Old replaced by text New (when searching from the left). If Which is omitted, every occurrence of Old is replaced with New; if Which is provided, only that occurrence of Old is replaced by New (starting the count from 1). If there is no match, or if Old has length 0, the value of T is returned. Note that Old and New may have different lengths. If Which is present and Which < 1, returns Error.

See also LEFT 6.20.12, LEN 6.20.13, MID 6.20.15, REPLACE 6.20.17, RIGHT 6.20.19

6.20.22 T

Summary: Return the text (if Text), else return 0-length Text value

Syntax: T( Any X )

Semantics: The type of (a dereferenced) X is examined; if it is of type Text, it is returned, else an empty string (Text value of zero length) is returned. This is not a type-conversion function; T(5) produces an empty string, not "5".

6.20.23 TEXT

Summary: Return the value converted to a text.

Syntax: TEXT( Scalar X ; Text FormatCode )

Constraints: The FormatCode is a sequence of characters with an implementation-defined meaning.

Semantics: Converts the value X to a Text according to the rules of a number format code passed as FormatCode and returns it.

6.20.24 TRIM

Summary: Remove leading and trailing spaces, and replace all internal multiple spaces with a single space.

Syntax: TRIM( Text T )

Semantics: Takes T and removes all leading and trailing space. Any other sequence of 2 or more spaces is replaced with a single space.

A space is one or more, HORIZONTAL TABULATION (U+0009), LINE FEED (U+000A), CARRIAGE RETURN (U+000D) or SPACE (U+0020) characters.

6.20.25 UNICHAR

Summary: Return the character represented by the given numeric value according to the [UNICODE] Standard.

Syntax: UNICHAR( Integer N )

Returns: Text

Constraints: N ≥ 0, N ≤ 1114111 (U+10FFFF)

Semantics: Returns the character having the given numeric value as [UNICODE] code point.
Evaluators shall support values between 1 and 0xFFFF. Evaluators should allow N to be any [UNICODE] code point of type Graphic, Format or Control. Evaluators should implement UNICHAR such that UNICODE(UNICHAR(N)) returns N for any [UNICODE] code point N of type Graphic, Format or Control.

6.20.26 UNICODE

Summary: Return the [UNICODE] code point corresponding to the first character of the text value.

Syntax: UNICODE( Text T )

Returns: Number

Constraints: Length(T) > 0.

Semantics: Returns the numeric value of the [UNICODE] code point of the first character of the given text T.

The results of this function may be normalization-sensitive. 4.2

6.20.27 UPPER

Summary: Return input string, but with all lowercase letters converted to uppercase letters.

Syntax: UPPER( Text T )